Structural Accounts ofMathematical Representation

David Andrew Race

Submitted in accordance with the requirements for the degree of

Doctor of Philosophy

The University of Leeds

The School of Philosophy, Religion & History of Science

October, 2014

The candidate confirms that the work submitted is his own and that

appropriate credit has been given where reference has been made to the work

of others.

This copy has been supplied on the understanding that it is copyright

material and that no quotation from the thesis may be published without

proper acknowledgement.

© 2014 The University of Leeds and David Race

i

Acknowledgements

First I would to thank my supervisors, Steven French and Juha Saatsi. They

have provided valuable support, guidance and constructive criticism as needed.

Their patience in reading my many, lengthy and half finished drafts has been

fantastic. I have followed their help as much as I have been able to, but I

certainly have more to learn. I also want to thank the rest of the department at

the University of Leeds. In particular Nicholas Jones for his constant support in

all matters related to teaching; Joseph Melia and Paolo Santorio for allowing

me to lecture on their units and the valuable feedback they gave; and to

Jonathan Topham and Graeme Gooday for their help and encouragement in

organising and finding funding for the White Rose Philosophy Postgraduate

Forum. Thanks also to everyone who attended the reading groups I joined

and the presentations I gave. The discussions and feedback were always useful

and helped broaden my horizons. I am also grateful to the University of York,

in particular Mary Leng, for offering me the chance to lead their third year

Philosophy of Maths unit. Thank you to the Arts and Humanities Research

Council who provided generous funding for my research.

This thesis could never have been completed without the friendships I’ve

made here at Leeds. A huge thank you to all of you, especially those of

you in the department: Jon Banks, Becky Bowd, Efram Sera-Shirar, Thomas

Brouwer, Carl Warom, Jordan Bartol, Richard Caves, Michael Bench-Capon,

Dani Adams, Sarah Adams, Wouter Kalf, Kerry McKenzie, Jamie Stark, Dom

ii

Berry, and anyone else who’s slipped my mind. You’ve all helped, through

discussing serious (and not serious) philosophical arguments to just being there

to share a drink with. Thank you to my friends outside of the department,

who provided a much needed dose of normality: Andy Ness, Dan Horner, Tim

Stock, Sam Grant, JJ Coates, Mungo Dalglish, Luke Bowen, Seb Atkinson

and many more. Thanks to my friends from Bristol and home. This thesis

prevented me from seeing you as much as I’d have liked, but it never felt like

I’d been away when I did get to see you. The feeling of lasting friendship was

an important motivator to finish.

Finally, and most of all, I’d like to thank my family for the continued belief

and support. Thank you for the unwavering encouragement in both me, and

my ability to complete this work. To Ian and Sue, this is for you.

iii

Abstract

Attempts to solve the problems of the applicability of mathematics have gen-

erally originated from the acceptance of a particular mathematical ontology.

In this thesis I argue that a proper approach to solving these problems comes

from an ‘application first’ approach. If one attempts to form the problems and

answer them from a position that is agnostic towards mathematical ontology,

the difficulties surrounding these problems fall away. I argue that there are

nine problems that require answering, and that the problems of representation

are the most interesting questions to answer. The applied metaphysical prob-

lem can be answered by structural relations, which are adopted as the starting

point for accounts of representation. The majority of the thesis concerns ar-

guing in favour of structural accounts of representation, in particular deciding

between the Inferential Conception of the Applicability of Mathematics and

Pincock’s Mapping Account. Through the case study of the rainbow, I argue

that the Inferential Conception is the more viable account. It is capable of

answering all of the problems of the applicability of mathematics, while the

methodology adopted by Pincock trivialises the answer it can supply to the

vital question of how the faithfulness and usefulness of representations are

related.

iv

Contents

1 Introduction 1

2 The Philosophical Problems of the Applicability of Mathemat-

ics 8

2.1 Introduction: What is the Applicability of Mathematics? . . . . 8

2.2 Examples of Applied Mathematics . . . . . . . . . . . . . . . . . 13

2.2.1 SU(3) and Sub-Atomic Physics . . . . . . . . . . . . . . . 13

2.2.2 Dirac and the Positron . . . . . . . . . . . . . . . . . . . . 16

2.3 The Development of the Philosophical Problems . . . . . . . . . 21

2.3.1 The Metaphysical Problems . . . . . . . . . . . . . . . . . 22

2.3.2 Representation & The Descriptive Problem . . . . . . . . 23

2.3.3 Novel Predictions . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.4 Isolated Mathematics and Steiner’s Epistemological Prob-

lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.5 The Semantic Problem . . . . . . . . . . . . . . . . . . . . 38

2.3.6 The Mathematical Explanation Problem . . . . . . . . . 39

2.4 Solving the Applied Metaphysical Problem . . . . . . . . . . . . 41

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Representation 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 What is Representation? . . . . . . . . . . . . . . . . . . . . . . . 48

v

3.3 A Special Problem for Scientific Representation? . . . . . . . . . 49

3.3.1 The Constitution Question . . . . . . . . . . . . . . . . . 52

3.3.2 The Demarcation Question . . . . . . . . . . . . . . . . . 65

3.4 Contessa’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1 Necessary & Sufficient Conditions for Epistemic Repre-

sentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.2 Necessary & Sufficient Conditions for Surrogative Infer-

ences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.3 Faithful Epistemic Representation . . . . . . . . . . . . . 76

3.4.4 Scientific and Epistemic Representation . . . . . . . . . . 76

3.5 Chakravartty: Informational vs. Functional Accounts of Repre-

sentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.6 Conclusion: Answering Callender and Cohen’s Questions . . . . 85

3.6.1 Answering the Constitutive Question . . . . . . . . . . . 85

3.6.2 Answering the Normative Issue . . . . . . . . . . . . . . . 86

4 Structural Relation Accounts of Representation 88

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2 Structural Mapping Accounts of Representation . . . . . . . . . 90

4.2.1 The Inferential Conception of the Applicability of Math-

ematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2.2 Pincock’s Mapping Account . . . . . . . . . . . . . . . . . 99

4.3 The Structural Mapping Accounts & Epistemic Representation 113

4.3.1 The Inferential Conception . . . . . . . . . . . . . . . . . 115

4.3.2 The Mapping Account . . . . . . . . . . . . . . . . . . . . 119

4.4 The Structural Mapping Accounts & Faithful Epistemic Repre-

sentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.4.1 The Inferential Conception . . . . . . . . . . . . . . . . . 131

4.4.2 The Mapping Account . . . . . . . . . . . . . . . . . . . . 133

vi

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5 The Rainbow: An Introduction 137

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2 The Ray Theoretic Approach to The Rainbow Angle and Colours139

5.3 The Mie Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.4 Supernumerary Bows & The CAM Approach . . . . . . . . . . . 145

5.5 Philosophical Responses . . . . . . . . . . . . . . . . . . . . . . . . 153

6 The Inferential Conception and the Rainbow 156

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2 The Inferential Conception and Idealisation . . . . . . . . . . . . 158

6.2.1 Partial Truth and Idealisation . . . . . . . . . . . . . . . . 159

6.2.2 Partial Structures and Idealisations . . . . . . . . . . . . 164

6.2.3 Surplus Structure on the Inferential Conception . . . . . 171

6.3 The IC Applied to the Rainbow . . . . . . . . . . . . . . . . . . . 181

6.3.1 Beta to Infinity Idealisation . . . . . . . . . . . . . . . . . 181

6.3.2 CAM Approach . . . . . . . . . . . . . . . . . . . . . . . . 186

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7 Pincock’s Mapping Account and the Rainbow 197

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.2 Pincock’s Mapping Account and Idealisation . . . . . . . . . . . 198

7.2.1 Account of Idealisation . . . . . . . . . . . . . . . . . . . . 198

7.2.2 PMA, Faithfulness and Usefulness . . . . . . . . . . . . . 204

7.3 The PMA Applied to the Rainbow . . . . . . . . . . . . . . . . . 221

7.3.1 Beta to Infinity Idealisation . . . . . . . . . . . . . . . . . 221

7.3.2 CAM Approach . . . . . . . . . . . . . . . . . . . . . . . . 226

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

vii

8 Choosing An Account 243

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.1.1 The Inferential Conception . . . . . . . . . . . . . . . . . 244

8.1.2 Pincock’s Mapping Account . . . . . . . . . . . . . . . . . 247

8.2 Are the Accounts of Representation Successful? . . . . . . . . . 250

8.2.1 The Isolated Mathematics Problem . . . . . . . . . . . . 251

8.2.2 Multiple Interpretations . . . . . . . . . . . . . . . . . . . 252

8.2.3 The Novel Predictions Problem . . . . . . . . . . . . . . . 254

8.2.4 The Problem of Misrepresentation . . . . . . . . . . . . . 259

8.3 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8.3.1 The Brading, Landry & French Debate . . . . . . . . . . 262

8.3.2 Pincock’s Methodological Approach . . . . . . . . . . . . 268

8.3.3 Accounting For Pincock’s Epistemic Contributions . . . 271

8.4 Choosing An Account . . . . . . . . . . . . . . . . . . . . . . . . . 276

8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9 Conclusion 281

A Appendix: Mathematics of the Rainbow 288

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

A.2 Mie Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 289

A.3 Debye Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

A.4 Poisson Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

A.5 Unified CAM Approach . . . . . . . . . . . . . . . . . . . . . . . . 311

A.5.1 Third Debye Term . . . . . . . . . . . . . . . . . . . . . . . 312

A.5.2 CFU Method for Saddle Points . . . . . . . . . . . . . . . 313

A.6 Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Bibliography 318

viii

List of Figures

2.1 Charge verses Strangeness plot of the heavy Baryons . . . . . . 15

4.1 Schematic diagram of the Inferential Conception . . . . . . . . . 95

5.1 Basic geometry of a ray incident a spherical water drop . . . . . 139

5.2 Diagram demonstrating that light groups around the minimum

angle of deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.3 Plot of Regge poles . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.4 Plot of Regge-Debye poles . . . . . . . . . . . . . . . . . . . . . . 151

1

1 | Introduction

The relationship between mathematics and science is one of the most active

areas of research in contemporary philosophy of science. The issues discussed

range from the general, such as whether mathematics is indispensable to sci-

ence, to the specific, such as whether prime numbers are explanatory. This

thesis will look at the general problem of the applicability of mathematics.

This problem can be initially characterised as the question “how is mathe-

matics applicable to the world?” A little reflection, however, shows that this

initial characterisation is seriously underdeveloped; what is meant by “applica-

ble” is vague. One might be asking how a particular ontology of mathematics

can be related to the world. Alternatively, one might be wondering whether

mathematics is able to explain physical phenomena. Steiner (1998) noticed

the ambiguity in this question and attempted to resolve it through philosoph-

ical analysis. He argued that one could distinguish several different problems

that fall under the general problem of the applicability of mathematics, and

that confusing and conflating these problems leads to difficulty in providing

adequate answers to the issues.

I will argue (in Chapter 2) that part of the difficulty in answering these

questions originates in the adoption of an ‘ontology first’ approach to the

questions. It has typically been the case that philosophers have a particular

ontology of mathematics in mind when they set out what they consider to be

‘the’ problem (or problems) of applicability. For example, a Platonist about

2

mathematics is often challenged to provide an account of how abstract acausal

objects are related to the world, such that they can be used to gain knowl-

edge about the world. That is, to explain the role of mathematics in science.

Alternatively, nominalists are challenged to provide an account of how false

statements can be used to gain knowledge. I reject this approach and argue

in favour of an ‘applications first’ approach: one should attempt to answer

the problems of the applicability of mathematics in an ontologically neutral

fashion.

My aim for this thesis is to identify a philosophical account of the appli-

cability of mathematics that will answer the relevant problems. My approach

to identifying such an account will be to properly distinguish the numerous

problems that fall under the umbrella of ‘the applicability of mathematics’,

argue that some of these problems depend on answers to others, and show

how my favoured account can answer these problems. Specifically, I will ar-

gue that the ‘applied metaphysical problem’ should be answered first. This

problem asks for a link between the mathematical and physical domains and

little more. I will also argue that the more interesting and difficult question

to answer is the problem of representation, the question of how mathematics

is able to represent the world. Several of the problems of applicability can

be understood as being (sub)problems of the problem of representation: is-

sues related to misrepresentation, and the issue of how multiple (sometimes

conflicting) interpretations of the same mathematics can each be partially suc-

cessful. As I consider the problems of representation to be the most difficult

of the problems to solve, the majority of this thesis will focus on issues related

to scientific representation.

In Chapter 2 I will set out nine distinct problems that fall under the um-

brella of ‘the applicability of mathematics’. These nine problems are:

1. The pure metaphysical problem.

3

2. The applied metaphysical problem.

3. The problems of representation:

3.a) The problem of representation.

3.b) The problems due to misrepresentation.

3.c) The multiple interpretations problem.

4. The novel predictions problem.

5. The problem of isolated mathematics.

6. The semantic problem.

7. The mathematical explanation problem.

I will follow Steiner’s (1998) and Pincock’s (2012) development and analysis

of these problems. I will argue that the pure metaphysical problem should be

answered by philosophers of mathematics, and that the semantic and mathe-

matical explanation problems require far more work to be undertaken before

they can be answered. As such, I will not answer these problems in this the-

sis. I will answer the applied metaphysical problem by appealing to structural

relations. The main motivation for this answer is that the bar is set very low

for a satisfactory answer to the applied metaphysical problem, but not for an

account of representation. An adequate account of mathematical representa-

tion will have to fulfil certain criteria (discussed in Chapter 3). These criteria

will dictate what will be required of the representation relation, which relates

the mathematical and physical domains. As such, the representation relation

is (in part) the answer to the applied metaphysical question.

I will conduct my investigation of representation in Chapter 3. There I will

argue in favour of structural accounts of scientific representation. The chapter

is structured around the question of whether there is a unique feature of scien-

tific representation. I will adopt the position that representation is an activity

4

that has a purpose, and that the purpose can set criteria for the representation

relation. I will argue that scientific representation is most likely what Con-

tessa defines as partially faithful epistemic representation. A partially faithful

epistemic representation is one where our representational vehicle allows us

to draw at least one sound surrogative inference about the representational

target. An advantage of adopting structural accounts of representation is that

we are able to use the (formal) notions of structural similarity to ground the

notion of partial faithfulness. This, in turn, allows us to explain why represen-

tational vehicles which purposefully misrepresent their targets can be useful.

In Chapter 4 I will set out two structural accounts of representation: the In-

ferential Conception of the Applicability of Mathematics,1 and Pincock’s most

recent Mapping Account.2 In addition to introducing these accounts in this

chapter, I will also investigate how they fit into Contessa’s approach towards

representation, which involves distinguish epistemic and faithful epistemic rep-

resentation.3 I will argue that Pincock’s Mapping Account can be understood

as an account of epistemic representation, but the Inferential Conception can-

not. However, I will sketch how both can be understood as accounts of faithful

epistemic representation. I will only provide a sketch of this claim in Chapter

4, because establishing whether either account succeeds as an account of faith-

ful epistemic representation takes a lot of careful, detailed argument centred

around the relationship between the faithfulness and usefulness of representa-

tions. This argument will be undertaken in Chapters 6 and 7.

Before I conduct the investigation into the success of the accounts, I set out

my central case study, the rainbow, in Chapter 5. In this chapter I will outline

the core mathematical ideas behind three representations of the rainbow: the

1See Bueno & Colyvan (2011), Bueno & French (2011, 2012).2Pincock (2012).3Epistemic representation is characterised as a representation that is used to learn some-

thing about its target. It consists in the drawing of valid surrogative inferences. Faithfulepistemic representations concerns the soundness of those surrogative inferences. See Con-tessa (2007, 2011).

5

ray theoretic representation; the Mie representation; and the Complex Angu-

lar Moment (CAM) approach. These representations all involve various types

of idealisations, abstractions and mathematical techniques. Furthermore, they

are related to each other in interesting ways. With an understanding of the

rainbow in hand, I will turn to the arguments concerning the viability of the

Inferential Conception and Pincock’s Mapping Account being accounts of faith-

ful epistemic representation. I will address each account individually. My ap-

proach to each account will be the same: I will establish whether the account

can use the same structural resources to accommodate different types of ide-

alisation; and whether the account provides an adequate explanation for how

the faithfulness and usefulness of representations are related given the answer

to the first issue.4 During the analysis of the two accounts, I will employ the

β →∞ singular limit, which is used to obtain a ray representation from wave

theory, and the abstract mathematics involved in the CAM approach. These

features of the representations of the rainbow, what role they play, how they

are obtained, and so on will form crucial test cases for the conclusions I draw

in answering the above two questions.

In Chapter 6 I will discuss the Inferential Conception. I will identify partial

structures and the associated notion of partial truth as being capable of pro-

viding an explanation of how partially faithful representations can be useful,

and that the same structural resources can be used for Galilean and singular

limit idealisations. However, I will identify issues concerning how idealisations

and the notion of ‘surplus structure’ should be accommodated on the Inferen-

tial Conception. As the Inferential Conception is part of the partial structures

version of the Semantic View, other work on this topic can be appealed to.

I will identify a dilemma generated by the attempt to accommodate surplus

4I will focus on two different types of idealisation: Galilean and singular limit idealisa-tions. See McMullin (1985) for Galilean idealisations, Batterman (2001) for singular limitidealisations. See also Weisberg (2007, 2013) for a different way of distinguishing these typesof idealisation.

6

structure on the Inferential Conception. I will employ the β →∞ limit to test

the conclusions concerning idealisations and surplus structure. I will use the

mathematics of the CAM approach to test the solutions to the dilemma. In

doing so I generate a further problem due to the possible indispensability of ray

theoretic concepts in the creation of the CAM approach models. I will argue

that a possible solution to this problem is to regard the Inferential Conception

as a third ‘axis’ of the Semantic View, to compliment the horizontal axis of

inter-theory relations and the vertical axis of data, phenomena and abstract

model relations. The Inferential Conception passes these tests, though further

work is required to flesh out the solutions.

I will discuss Pincock’s Mapping Account in Chapter 7. I will argue that it

requires different structural resources to accommodate Galilean idealisations

compared to singular limit idealisations. Pincock argues for understanding

some instances of singular limit idealisations in terms of singular perturba-

tions. As part of this argument, he advocates an interpretative position of

‘metaphysical agnosticism’ towards these perturbations, rejecting instrumen-

talist and metaphysical (i.e. emergentist or reductionist) positions. The Map-

ping Account appears to be capable of accommodating the relationship be-

tween faithfulness and usefulness for Galilean idealisations, through the use of

the specification relation and structural relation that relate the vehicle to the

target. I will find the case to be less clear for the singular limit (perturbation)

idealisations. I will argue that Pincock’s arguments in favour of his metaphys-

ical agnosticism are lacking, and his exposition of the position is unclear. I will

employ the β → ∞ idealisation and CAM approach in an attempt to better

understand metaphysical agnosticism, but I will conclude that it is either too

undeveloped a position to currently hold, or that it collapses into a form of

reductionism.

I will decide between the Inferential Conception and Pincock’s Mapping

7

Account in Chapter 8. To choose between the two accounts I will draw on the

conclusions of the previous chapters (6 and 7) to answer the relevant problems

of applicability from Chapter 2. These are the problem of isolated mathemat-

ics, the novel predictions and the multiple interpretations problem. Answering

the problem of representation and problems due to misrepresentation (specifi-

cally the issues raises by abstraction and idealisation) requires explaining how

an account of representation can provide faithful epistemic representation; as

such, the conclusion to Chapter 7 means that I will not able to immediately

answer these problems for Pincock’s Mapping Account. I will be able to an-

swer them for the Inferential Conception. Before concluding in favour of the

Inferential Conception, however, I will conduct a short investigation into the

methodology adopted by Pincock to construct his account. I will draw on the

analogous argument between French and Brading & Landry over the role of

the philosopher of science. The root of their disagreement is a rejection by

Brading & Landry of the ‘meta-level’ that French claims the philosopher of

science operates at. I will argue that Pincock’s position is similar to Brading

& Landry’s, such that he blurs the distinction between the meta-level and the

object level. This results in a trivialisation of the relationship between the

faithfulness and usefulness of representations. Due to this trivialisation, I will

reject Pincock’s Account, and adopt the Inferential Conception as the most

plausible account of representation.

8

2 | The Philosophical Problems of

the Applicability of

Mathematics

2.1 Introduction: What is the Applicability of

Mathematics?

In this chapter I will argue that the problem of the applicability of mathematics

is really a collection of problems, such as the pure and applied metaphysical

problems, the problems of representation and the mathematical explanation

problem. I argue that the applied metaphysical problem, the problems of

representation, the problem of novel predictions and the problem of isolated

mathematics to be the problems we should aim to answer first. This chapter

is concerned with explicating these problems, and justifying why I think these

problems are the ones which should be answered first.

I will present two examples of how mathematics can be applied to the

physical world: the use of the SU(3) group to classify the heavy baryons and

predict the quark model; and Dirac’s relativistic wave equation and the pre-

diction of the positron. I will use these examples to motivate some intuitive

worries about how applications work. I will then turn to one of the most im-

portant works on the applicability of mathematics, Steiner’s The Applicability

9

of Mathematics as a Philosophical Problem1 and contrast the intuitive worries

generated by the examples to the problems identified by Steiner. Some agree-

ment is found, though I will also argue against the way Steiner forms some

of his problems (mostly in line with Pincock’s criticisms (Pincock 2012, §8)).

This is due in part to the ontology first approach Steiner has adopted: Steiner

argues for a Fregean ontology of mathematics, which he accepts due to Frege’s

solution to the semantic problem that arises due to number terms acting as

predicates in some contexts and singular terms in others (Steiner 1998, pg.

16-17).2 I will end the chapter by arguing that the applied metaphysical prob-

lem is straightforwardly answered by structural relation accounts, and that the

more interesting problems are those concerning how mathematics is capable of

representing the world.

The general approach to the applicability of mathematics has been to as-

sume a particular ontology of mathematics and then attempt to explain how

that ontology either gets around problems that are due to that particular on-

tology3 or does not actually suffer from them.4 I take this approach to be a

consequence of indispensability arguments for mathematics. These arguments

claim that we should believe in those entities that are indispensable to our best

scientific theories and that mathematical entities are such objects, therefore we

should believe in the existence of mathematical entities.5 Pincock (2004b) ar-

gues that indispensability arguments rest on a misunderstanding of how math-

1Steiner (1998).2From this Fregean ontology, Steiner goes on to argue that the problems of applicability

have consequences for the rationality of pursuing naturalistic approaches to science. I willnot assess the arguments and claims concerning naturalistic approaches to science as thereis already a body of literature which argues against the conclusions Steiner draws againstnaturalism. See Simons (2001), Liston (2000), Kattau (2001), Bangu (2006) and Pincock(2012). I will however introduce the arguments in passing, as what Steiner has to sayabout the “Pythagorisation” of prediction is tied to how he sets up problems related to novelprediction and “the unreasonable effectiveness of mathematics” Wigner (1960).

3e.g. How are the non-spatiotemporal, acausal abstract objects of mathematical Platon-ism related to the world in an informative way?

4e.g. That some anti-realist positions hold mathematical statements to be literally falseis not a problem for accounting for the applicability of mathematics

5See Colyvan (2001).

10

ematics can be applied to the world and so draw unwarranted conclusions.6

This disagreement over how applications work and what the indispensability

argument shows, allows one to disagree with Colyvan over whether applica-

tions dictate mathematical ontology. I agree with Pincock’s evaluation of these

arguments and buy into his project of attempting to understand applications

of mathematics before attempting to draw any ontological commitments.

This line of argument is similar to the one adopted by Yablo (2005). He

argues that from a Fregean perspective towards mathematics (i.e. that it is a

science of abstract, sui generis objects) applicability can be seen as either a

datum, where “the question is, what lessons are to be drawn from it?”, or “as a

puzzle, and the question is, what explains it, how does it work?” (Yablo 2005,

pg. 98). Yablo argues that the datum perspective is usually given priority,

which leads to the following line of argument (Yablo 2005, pg. 89-90):

[that] since applicability would be a miracle if the mathematics involved

were not true, it is evidence that mathematics is true. The . . . ap-

plicability [of mathematics] is [then] explained in part by truth. It is

admitted, of course, that truth is not the full explanation. But the as-

sumption appears to be that any further considerations will be specific

to the mathematics involved and the application. The most that can be

said in general about why mathematics applies is that it is true.

Attention then turns to questions of what makes mathematical claims true,

that is, what the appropriate ontology of mathematics is (Platonic forms,

structures, etc.). Yablo then proceeds to turn these perspectives around, and

attempts to address the puzzle perspective of applicability first.

I can draw some support from other anti-realists for this approach, in that

they provide accounts of how to understand applicability without reference to

mathematical entities. Their motivations for constructing such arguments are6Pincock argues that Colyvan’s indispensability argument only establishes the semantic

realism of applied mathematics - that such statements must have a truth value.

11

obvious, though different to mine. I am appealing to anti-realist arguments

that proceed in two stages after putting forward the hypothesis that mathemat-

ical entities do not exist. The anti-realists first have to show how applications

can be understood without reference to mathematical entities. Second, they

argue from such an understanding of applications that mathematical entities

do not exist. For example, Maddy’s Arealist (similar to a Formalist), argues

that the existence of abstract objects adds little to the utility of applied mathe-

matics (Maddy 2007, pg. 380-381); and Bueno (2005) argues that mathematics

can play a heuristic role and that it is the physics which is important in ap-

plications.7 Further work has to be done from these positions to argue that

mathematical entities themselves do not exist, as they are all compatible with

mathematical entities either existing or not. I only require the first part of

these arguments as they demonstrate that positing mathematical entities is

not necessary for understanding applications of mathematics.8

I therefore conclude that the best approach to explaining how mathemat-

ics is applicable is to adopt accounts that only provide answers to what I will

call the applied metaphysical problem, and remain neutral with respect to

what I will call the pure metaphysical problem. Thus I will endorse answers

to the applied metaphysical problem which are “independent of pure mathe-

matics” (Pincock 2004a, pg. 130). I call this approach the ‘application first’

approach, to contrast with the previous approaches, which I call ‘ontology first’

approaches.

I also appeal to scientific practice as support for the ‘applicability first’

7Bueno’s position towards mathematics is variable. He offers a first step towards a con-structive empiricist approach to mathematics in his (1999), which would consist in arguingfor an agnostic attitude towards mathematical entities. In his (2009) he argues in favour ofmathematical Fictionalism.

8I have not appealed to Field here as I follow Yablo’s argument that just as the applica-bility of mathematics can be see as an argument or a problem, so too can indispensability,and that Field focuses on indispensability rather than applicability (Yablo 2005, pg. 92).The question that Yablo and I am concerned with is “how are actual applications to beunderstood, be the objects indispensable or not?”

12

approach. Applications of mathematics by scientists do not, in general, make

any reference to the ontology of the mathematics used.9 This is the case in

the examples below. This suggests that applications of mathematics do not

dictate that a side has to be taken on the realism/anti-realism debate in math-

ematics. As this claim is in direct opposition to Colyvan’s position, remember

that Pincock rejects Colyvan’s indispensability argument, arguing that it only

establishes semantic realism about mathematics. Pincock claims that Coly-

van’s argument should be reformulated with a weaker premise of “apparent

reference to mathematical entities is indispensable to our best scientific the-

ories” (Pincock 2004b, pg. 68). I therefore suggest (but will not argue) that

applications, at most, restrict the range of possible (particular) mathematical

ontologies. That is, while applications may rule out individual theories (Pla-

tonic forms or neo-Aristotelianism for example), they do not force one to be a

realist (or anti-realist) about mathematics. Of course, all anti-realist strategies

might be ruled out. However, such a situation would be a failure of anti-realist

accounts individually as opposed to anti-realism being categorically ruled out.

With regards to mathematical ontology, the mappings will require that the

mathematical domain displays certain (structural) features. This restrictive

move is similar to that taken by Bueno (2009), who argues that an ontology of

mathematics must fulfil certain criteria to be viable, explicitly discussing the

applicability of mathematics.

Now I will outline the two examples of applied mathematics: the applica-

tion of the SU(3) group to quarks; and the Dirac equation and the prediction

of the positron.

9Colyvan challenges this point via the example of the imaginary number i. He arguesthat mathematicians did not accept it as a number until after Gauss made use of it in hisproof of the fundamental theorems of algebra and physical applications for complex functiontheory were found (Colyvan 2002, pg. 104-105). I tentatively suggest that this shows thatscientists are unconcerned about the ontological status, given that applications had to befound before i was accepted by mathematicians.

13

2.2 Examples of Applied Mathematics

2.2.1 SU(3) and Sub-Atomic Physics

In the 1950s, there were a large number of new, supposedly fundamental,

particles being discovered. Besides the proton, neutron and electron, particle

physicists were aware of the pions (π+, π0, π−), kaons (K+,K0,K−) and eta

(η) mesons, and the lambda (Λ), delta (∆++,∆+,∆0,∆−), sigma (Σ+,Σ0,Σ−)

and xi (Ξ+,Ξ−) baryons. Establishing which decay modes were allowable and

which were not was a major problem caused by this wide range of particles.

A similar problem had arisen in the 1930s with the proton, when the question

was asked as to why it did not decay. A solution was found by requiring a

new property to be conserved, the baryon number. These new particles were

proposed to display a further new property, that of ‘strangeness’.10

An attempt to categorise these new particles was made by Gell-Mann. He

started by looking at isospin, an approximate symmetry that can be found

between protons and neutrons.11 The idealisation used to construct isospin

allowed for various calculations and predictions to be simplified. Gell-Mann

attempted to generalise the isospin strategy to the new particles by searching

for symmetries for the strong, weak and electromagnetic forces (Gell-Mann

1987, pg. 483-489).

Isospin demonstrates the roles that idealisation and abstraction play in

science and hint at how mathematics facilitates these processes. Following the

distinction drawn by Jones, an abstraction is the omission of detail by means

10The particles were considered ‘strange’ as they had a short creation time and a (rel-atively) slow decay time, indicating a different process was involved in their creation totheir decay. In fact, strong interactions (the interaction that holds the nucleus together)conserve strangeness and are responsible for the creation of strange particles, whereas weakinteractions (the interaction responsible for beta decay) are responsible for the decay of theparticles and do not converse strangeness.

11It was observed that these two particles are almost identical if their charges are ignored.It was therefore proposed to consider them as two different states of the same particle, a‘nucleon’.

14

of saying nothing about it, while idealisations are the inclusion of a falsehood

about a system (Jones 2005, pg. 175). In the isospin case, the equating of the

masses of the proton and neutron is an idealisation, while the lack of detail

on the charge of the neutron or proton is an abstraction. We need to account

for how mathematics is able to provide the tools for these moves and for how

making the moves can be justified.

It was pointed out to Gell-Mann that the algebra he was using is actually

the Lie algebra of the group SU(3). A group is a set of elements together

with an operator, such that the operator combines any two elements into a

third element.12 Groups can be represented by matrices. The matrices used

to represent a group can be generated from the Lie algebra. Arranging the

nine known heavy baryons (∆,Σ∗,Ξ∗, where an excited state is notified by

a * superscript) according to their strangeness and charge as in Figure 2.1

produces a geometry which is almost identical to one found in a particular

representation of the SU(3) group. This representation gives a decuplet, an

arrangement of 10 points. Due to experimental results which allowed this

representation to be made, Gell-Mann was able to predict the existence of a

10th heavy baryon, the Ω−, and its mass (Gell-Mann 1987, pg. 492). The light

baryons form an octet when plotted in the same way, while the mesons produce

a hexagon, each of which is identical to the geometry of other representations

of SU(3).

The prediction of the Ω− is a case of novel prediction, a key test for a

new theory. There are two features of this prediction that are of particular

interest and involve the use of mathematics. First, the prediction is not made

in the usual way as being the result of anomalous interactions that require

explanation.13 Rather, the Ω− was “read off” from the mathematics. Second,

12See Georgi (1999) Chapter 1 for details of what constitutes a group and Chapters 7and 9 for information on SU(3).

13See Bangu (2006) and §2.3.3 below for more on this claim.

15

34 1/HISTORICAL INTRODUCTION TO THE ELEMENTARY PARTICLES

charge) for the proton and the Q = 0 for the neutron, the lambda, the and the EO; Q = -1 for the and the E-. Horizontal lines associate particles of like strangeness: S = 0 for the proton and neutron, S = -1 for the middle line and S = -2 for the two Z's.

The eight lightest mesons fill a similar hexagonal pattern, forming the (pseudo-scalar) meson octet:

\ \ \ \

\ \

0=-1 0=0

\ \

\ \

\ \

\

0=1

The Meson Octet

Once again, diagonal lines determine charge, and horizontals determine strange-ness; but this time the top line has S = 1, the middle line S = 0, and the bottom line S = -1. (This discrepancy is a historical accident; Gell-Mann could just as well have assigned S = 1 to the proton and neutron, S = 0 to the and the A, and S = -1 to the E's. In 1953 he had no reason to prefer that choice, and it seemed most natural to give the familiar particles-proton, neutron, and pion-a strangeness of zero. After 1961 a new term-hypercharge-was introduced, which was equal to S for the mesons and to S + 1 for the baryons. But later developments showed that strangeness was the better quantity after all, and the word "hypercharge" has now been taken over for a quite different purpose.)

Hexagons were not the only figures allowed by the Eightfold Way; there was also, for example, a triangular array, incorporating 10 heavier baryons-the baryon decuplet:

A-S = 0 - - - - __ ---------....,\

\ \

*+ \

\ \

\ 0=-1

\

'" \ \ 0=2 \

\

\ 0=0

\ \

0=1

The Baryon Decuplet

Figure 2.1: Charge verses Strangeness plot of the heavy Baryons, including Ω−.Reproduced from Griffiths (1987).

the mathematics that is used to produce the prediction is highly abstract.

SU(3) does not describe physical rotations as is the case with other groups.

For example SU(2) can be used to describe angular momentum symmetry.

All symmetries that are found in a SU(n) group occur in some m-dimensional

space, the dimension of which is determined by the following equation: m =

n2 − 1. Therefore, SU(2) rotations occur in a 3-dimensional space, whereas

SU(3) rotations occur in an 8-dimensional space. This raises the question as

to why a piece of mathematics that describes rotations in an 8-dimensional

space is appropriate for describing a collection of particles. How are we able

to generate novel predictions by operations on such abstract mathematics?

The fundamental representation of the SU(3) group produces the geometry

of a equilateral triangle. This representation allowed Gell-Mann to draw the in-

ference that hadrons could be understood as being composed of a combination

of three fundamental parts. He did not originally consider these particles to be

‘real’, only ‘mathematical’, in that he did not think it was possible to observe

the parts in isolation and so would be permanently confined within hadrons

(Gell-Mann 1987, pg. 493). These parts are the foundation of the quark theory;

specifically, they are the light quarks: up, down, and strange. Quarks have

fractional charge, fractional baryon number and the strange quark carries a

16

strange quantum number of −1. Baryons are composed of a combination of

three quarks or antiquarks, while mesons are composed of a quark-antiquark

pair.

SU(3)’s contribution towards quark theory highlights a further role that

mathematics appears to be able to play, that of explanation. It seems that

answers to the questions raised above can be found in the mathematics itself;

that the symmetries of the physical system being described are completely

reproduced in the symmetries of SU(3) (given some abstractions and idealisa-

tions). The mathematics is therefore able to “explain” why we observe particu-

lar hadrons (only certain combinations of quarks are allowable, or only certain

symmetries can be obtained with 3 fundamental constituents, for example).

2.2.2 Dirac and the Positron

Dirac’s prediction of the positron was not a straightforward affair. A negative

energy solution was obtained from his relativistic wave equation. This solution

required interpretation within Quantum Mechanics. It was interpreted several

times, as several of the interpretations failed tests of empirical adequacy. I will

summarise the derivation of the Dirac equation and the relevant result. Then

I will briefly discuss the inadequate interpretations and the one that predicted

the positron as we now understand it.

The starting point for Dirac’s attempts to find a relativistic quantum theory

was the time dependent Schrödinger wave equation, (2.1). The Schrödinger

equation needs to be manipulated to work with relativistic energies, masses

and momentums. I will now present the derivation of the Dirac equation for a

free particle of spin 12 .

14

14The derivation is taken from Rae (2008).

17

−ih∂Ψ

∂t= HΨ (2.1)

E2 = p2c2 +m2c4 (2.2)

Assume that the energy operator, H, can be expressed in terms of the

momentum operator, P , in the same way as the energy, E, and momentum, p,

are in the classical limit, i.e. as in (2.2), to give (2.3). This gives an operator

that is the square root of another operator. There is no clear way of how to

handle this situation, but to preserve Lorentz invariance, we need the position

and time coordinates to be of the same order in a relativistic equation. The

left hand side is linear in time, so the right hand side must have linear position

and time coordinates, as in (2.4).

−ih∂Ψ

∂t=√

[P 2c2 +m2c4]Ψ (2.3)

−ih∂Ψ

∂t= [cγ1Px + cγ2Py + cγ3Pz + γ0mc

2]Ψ (2.4)

In order to resolve the equality and the relations required of the γi coeffi-

cients, the coefficients cannot be scalar numbers; the most simple expressions

for them are a set of 4 by 4 matrices. The final Dirac equation is given by (2.5).

The use of matrices in the Dirac equation requires that the two component

wave function, Ψ, of the Schrödinger equation is replaced with a four element

“spinor", or column matrix, ψ, as in (2.6).

18

ihγµ∂µψ −mcψ = 0 (2.5)

ψ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ψ1

ψ2

ψ3

ψ4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=⎛⎜⎜⎝

ψA

ψB

⎞⎟⎟⎠

(2.6)

The classical relativistic equation (2.2), has solutions for negative energy

values, less than −mc2. These are rejected as being unphysical (as are positive

values over mc2) in classical mechanics. Such a move cannot be made in

Quantum Mechanics however, as transitions to negative energy states are in

principle achievable (Rae 2008, pg. 255). ψB is the negative energy solution

for the Dirac equation and is the result from which the positron is ultimately

predicted.

The first interpretation of this solution by Dirac was an attempt to resist

a negative energy particle interpretation. He proposed that they were states

already filled by electrons. This is the famous Dirac sea of negative energy

electrons. Transitions to these states would then be prevented by the Pauli

exclusion principle.15 When one of these sea electrons is excited by a photon

with enough energy to move it to one of the positive energy states, a negative

energy state is left vacant - described as a ‘hole’. The excited electron behaves

as a normal electron would. The electron sea and hole, however, behave dif-

ferently. The total energy of the sea has increased (by the value of the photon

minus the mass of the electron), making their net momentum −p, where p is

the momentum of the excited electron. This momentum behaves as an electron

with a positive charge would in an electric field.

15The Pauli exclusion principle states that no two fermions may occupy the same quantumstate simultaneously.

19

Despite several disagreements with experimental results, Dirac’s equation

was well supported (Kragh 1990, pg. 65). This was due in part to agreement

with other formal results (Kragh 1990, pg. 63). The main complaints came

from the interpretation of the negative energy states. While formal scattering

results showed that the negative energy states had to be taken seriously (Kragh

1990, pg. 89), the interpretation of them as an unobservable sea of electrons

was resisted. Pauli objected to the “zero charge” of the sea for example (Pais

et al. 1998, pg. 52-53). Dirac attempted to interpret the hole as a particle,

while maintaining the sea of negative states: first as a proton, then, due to

objections by Heisenberg that this would require the mass of the proton and

electron to be equated, as the positron (Kragh 1990, pg. 94, 103). Empirical

confirmation came first from Anderson, who discovered a particle of similar

mass to an electron but positively charged from cosmic ray interactions in a

cloud chamber. Blackett and Occhialini confirmed Anderson’s findings and

explicitly identified the particle with that of the positively charged electron

from Dirac’s equation (Kragh 1990, pg. 109). Attempts at interpreting the

Dirac equation as an empty vacuum state were led by Pauli and Weisskopf,

and Oppenheimer and Furry, though their own theories were also considered

inadequate (Kragh 1990, pg. 114). These theories were attempts to produce

Quantum Field Theories with an empty vacuum state. We are now able to

provide such an interpretation with modern Quantum Field Theories, such as

Quantum Electron Dynamics. The Dirac equation is taken to describe the

Dirac field, which has the (empty) vacuum as its ground state. The field

is interpreted as being excited or de-excited which leads to the creation of

electrons and positrons respectively.

The derivation of the relativistic wave equation is a normal application of

mathematics. The prediction of the positron, however, is unusual. It raises

questions over the nature of interpretation: how can one piece of mathematics

20

have two very different interpretations yet both have some predictive success?

This seems to imply that the mathematics refers to something in the world

that the first interpretation did not recognise but the second did. The question

of how mathematics can provide novel predictions is again raised.

I will use these two case studies to develop and distinguish some of the prob-

lems of the applicability of mathematics. Steiner also made use of these two

case studies when he developed his set of problems, so they provide a common

ground from which begin. The SU(3) example is used by Steiner to generate

his descriptive problem, which is motivated by the idea that mathematics has

to be appropriate in some way for its application. I will argue, however, that

the abstractions and idealisations involved in the development of isospin show

that Steiner has missed several features of representation. I will subsequently

argue that we should reject Steiner’s descriptive problem in favour of the more

nuanced problems of representation. I will also use the SU(3) example, and

other uses that SU(3) has been put to, to clarify the problem of isolated

mathematics, which concerns whether mathematics can be applicable if it was

created or discovered in isolation from considerations of applicability. I will

use the prediction of the positron to argue that Steiner has missed another

problem of representation, that of multiple interpretations. Specifically, that

the negative energy solutions could have useful but inconsistent multiple inter-

pretations needs to be accommodated. Both case studies involve prediction;

Steiner argues that the prediction of the Ω− is a Pythagorean prediction, where

the prediction involves reifying the mathematics. I will use these case studies

to develop the problem of novel predictions, and later compare them to cases

of typical prediction when answering these problems.

I will now move on to setting out what problems I believe an account of

applicability must answer. I first argue that there is a separate metaphysical

21

problem for applied mathematics than for pure mathematics. This is a problem

initially identified by Steiner, and expanded by Pincock. I then argue for five

more problems and contrast how I establish the problems with similar problems

identified by Steiner.

2.3 The Development of the Philosophical

Problems

In this section I will look at the problems posed by a statement of applied math-

ematics. The problems that I will outline are similar to those found by Steiner.

I will follow Pincock’s approach towards these questions, outlining how I will

aim answer some of the problems and reasons for dismissing others. First, I

will distinguish two types of metaphysical problem and argue that the applied

metaphysical problem is the one which requires an answer in this thesis. Sec-

ond, I will draw on the SU(3) example in a discussion over Steiner’s descriptive

problem and conclude that the idealisations and abstractions involved in the

example motivate an account of scientific representation.16 Steiner’s descrip-

tive problem only touches on representation, however. Third, I will discuss

novel predictions and Steiner’s epistemological problem. This discussion will

involve criticisms drawn from Bangu’s arguments, the outlining of the novel

predictions problem, and the forming of the problem of isolated mathematics

from Steiner’s comments on the historical origins of SU(3). Finally, I set out

the semantic and mathematical explanation problems, and argue that they16In this chapter I only outline the problem of representation and the problem of misrep-

resentation due to abstractions and idealisations. In §3.3.1 - The Argument from Misrep-resentation I will outline three further problems. These are: mistargeting due to mistakentarget; mistargeting due to non-existent targets; and misrepresentation due to empiricallyinadequate results. In §3.3.1 - The Argument from Misrepresentation I solve the problemof mistargeting due to mistaken target and argue that empirically inadequate results are ageneral problem for philosophy of science. In §3.5 I argue for a solution to the problem ofmistargeting due to non-existent targets. The problem of misrepresentation due to abstrac-tions and idealisations is discussed throughout this thesis, in the context of establishing anaccount of faithful epistemic representation.

22

require too much work to be answered in this thesis.

2.3.1 The Metaphysical Problems

There are two types of metaphysical problem involved in the applicability of

mathematics that need to be distinguished. The first is the problem of what

mathematics refers to, asking if there are mathematical entities. I shall refer

to it as the pure metaphysical problem. This is a question for philosophers of

mathematics. I shall call the second problem the applied metaphysical problem.

This problem arises because for “almost any interpretation of mathematics”

there is a “gap between the subject matter . . . and its applications” (Pincock

2012, pg. 172). For example, on Lewis’ part-whole relation interpretation of

mathematics, “the theory of hyperbolic differential equations is about some dis-

persed collection of mereological wholes and this intricately structured entity

seems just as extrinsic to fluid systems as any platonic entity". This prob-

lem was originally formulated by Steiner as being specific to Platonism about

mathematics. He held that there was a ‘metaphysical gap’ that “blocks any

nontrivial relation between mathematical and physical objects, contradicting

physics which presupposes such relations” (Steiner 1998, pg. 20). His com-

plaint rests upon an argument that because science admits only causal and

spatiotemporal relations, and abstract entities do not enter into such relation-

ships, on the Platonic view, physical theories are false (Steiner 1998, pg. 21).

I follow Pincock, however, in taking the applied metaphysical problem to be

a problem for (almost) all interpretations of mathematics.17 As it is such a

general problem, I take it to be independent of the ontology of mathematics.

These problems can be characterised as asking different questions about

the mathematical domain. The pure metaphysical problem asks what inhabits

17The one interpretation that might avoid this problem would be a nominalist one thatidentifies equations with the physical systems they are describing. In such a situation hy-perbolic differential equations are about fluid systems, if they’re about anything at all.

23

the domain, while the applied metaphysical problem asks how the mathemat-

ical domain is related to the physical world, irrespective of what inhabits the

mathematical domain. Solutions to the applied metaphysical problem should

therefore not dictate what mathematical entities are. That is not to say that

the solutions will be completely silent on what inhabits to the domain, how-

ever. Depending on how the relations are understood, they will set certain

limits on what can inhabit the mathematical domain. For example, if one

insisted that for a relation to exist, that relation’s relata must exist then Fic-

tionalist accounts of mathematics would be prima facíe ruled out.

Above, I endorsed an ‘applications first’ approach to the problems of appli-

cability, rather than the ‘ontology first’ approach most commonly found in the

literature. This ‘applications first’ approach means that I should attempt to

answer the applied metaphysical problem prior to settling the pure metaphys-

ical problem. In this thesis I will only be answering the applied metaphysical

question; as I said above, the pure metaphysical problem is one for philoso-

phers of mathematics, and my answer to the applied metaphysical problem

should be as silent as possible on the pure metaphysical problem.

2.3.2 Representation & The Descriptive Problem

Next, I turn to Steiner’s descriptive problem. I will argue that Steiner has

been lead astray by his Fregean ontology, and that he has posed a badly

formed question that, understood properly, concerns mathematics’ ability to

represent the world. I then set out some of the subproblems of the problems

of representation: issues related to misrepresentation (due to abstraction and

idealisation), and multiple interpretations. More specific concerns with repre-

sentation are set out in the next chapter, where I argue for adopting structural

relation accounts of representation. The discussion here leads into the next

problem Steiner discusses, the problem of novel prediction.

24

Steiner’s Descriptive Problem

Steiner’s descriptive problem can be loosely phrased as asking for an explana-

tion as to why certain mathematics is capable of describing certain physical

phenomena. Initially this seems very similar to asking how a certain piece of

mathematics is able to represent a physical situation. However, as will become

clear, Steiner is not working with a notion of ‘description’ that can be under-

stood as representation, due to the way his Fregean ontology causes him to

phrase his descriptive problem. Steiner’s solution to his descriptive problem is

to attempt to “[explain] in nominalistic language” why a piece of mathematics

is appropriate to describe a situation. The explanation Steiner provides con-

sists in attempting to pair the mathematical concept with a physical concept.

Steiner’s descriptive problem originates in his adoption of the Fregean ontol-

ogy; as such, the applications first approach allows one to dismiss the problem

as being badly formed. However, I will set out his problem, and Pincock’s

response, as it is a good example of how the ontology first approach causes

unnecessary complications that the application first approach can avoid.

The SU(3) example established an intuitive worry that is similar to Steiner’s

descriptive problem. The question was asked why the 8 dimensional rotations

described by SU(3) were appropriate for describing the hadrons. Steiner ar-

gues for a way of assuaging this worry in his account of how the application

of SU(3) should be understood. The solution comes from SU(3) providing an

account of how the hadrons came to be built out of quarks. The quarks are

‘found’ from SU(3) as follows (stated in general terms): “The group SU(n) has

two representations of n dimensions. The eigenvectors of the first can be taken

to be the state vectors of the n quarks. The second is conjugate to the first,

and its eigenvectors represent the antiquarks” (Hendry & Lichtenberg 1978,

pg. 1716). This means that, as SU(3) has a representation of 3 dimensions,

the number of eigenvectors of the first representation, 3, gives the number of

25

quarks, 3. These eigenvectors are written thus: up: ( 100); down: ( 0

10); strange:

( 001). The matching is between these eigenvectors, mathematical concepts, and

the concept of the quarks, which is linked to the physical objects themselves.

SU(3)’s success in answering the descriptive question at first appears to

be uncontroversial. An objection can be raised which, although it can be

answered in this case, can be generalised to call into question Steiner’s framing

of the descriptive problem. The objection runs as follows: SU(3) has only

one application, and that is to the light quarks. What if SU(3) is found

to have another application far removed from quarks? The matching then

cannot be between the eigenvectors and the quark concept. The reply to this

is that SU(3)’s success is not due to the quarks themselves, but that there

are 3 objects which display the symmetries that are found in SU(3). Any

subsequent application of SU(3) would have to be to 3 objects which display

these same symmetries.

The above objection to SU(3) can be generalised thus: “why is it that a

single piece of mathematics must be associated to the same physical concept

in every application?” Steiner has not argued for this being the case other

than requiring it to be a feature of a successful answer. Pincock takes up this

issue with Steiner: “he is assuming that when a given mathematical concept

functions descriptively in two representations, this concept must be describing

the same physical property in both cases” (Pincock 2012, pg. 177). Pincock

shows that applications of the same piece of mathematics do not always involve

the same features of the mathematics, never mind involving a consistent, single,

physical concept across all applications.18

Pincock has shown, at the very least, that Steiner requires an argument

for his presupposition that each applied piece of mathematics requires a single

18He makes use of examples that show that superposition does not require a superpositionof causes and that different features of analytic functions are used in different applicationsof these functions (Pincock 2012, pg. 177).

26

physical property consistent to all its applications. I agree with Pincock that

“the way Steiner frames his descriptive question . . . makes it harder to answer

than it should be and suggests a mystery where none exists”. The crux of

this objection is a disagreement with Steiner’s understanding of ‘description’.

He seems to confuse application with description, in that any application of

mathematics is ‘descriptive’ of some physical feature, whereas this is just not

the case (Pincock 2012, pg. 177). Rather, description should be understood as

representation. Representation is, loosely, taken to be the use of mathematics

to ‘describe’ and gain information, about the world.19 There are several key

features of representation that Steiner does not address. Chief among these are

idealisation and abstraction. As I will now argue, accounting for idealisation

and abstraction is another problem that should fall under the umbrella of the

problems of the applicability of mathematics: it is one of the problems of

mathematical representation.

Idealisation & Abstraction

Gell-Mann’s starting point for classifying the light hadrons was isospin. As

summarised above, the origin of isospin involved both idealisation and ab-

straction. How mathematics allows one to perform these actions needs to be

accounted for. I claim that any account of applicability (i.e. the answer to

the applied metaphysical problem) is too simple to handle idealisation and

abstraction, and that such accounts will need to be expanded to take account

of scientific representation.20

Abstraction is the omission of detail. This is demonstrated by isospin not

including details of the charge of the neutrons or protons. Idealisation involves

including known falsehoods in one’s description21 of a situation. In the isospin

19I hold out against using the jargon of representation until the next chapter.20This is in part because idealisation and abstraction are examples of misrepresentation.

Other instances of misrepresentation are discussed in §3.3.1.21I use ‘description’ here without intending to commit myself to any particular view

27

case this occurs when treating the mass of the neutrons and protons as being

of equal value.

The way that mathematics can facilitate an act of abstraction is easy to

understand: where an equation with 10 variables accurately describes every

feature of a situation, including less variables constitutes abstraction. There is

no obvious way in which one can provide a mathematical explanation for why,

or which, variables are dropped. These reasons are external to the mathemat-

ics, such as what is considered relevant, or not relevant, to the representation,

and so can or cannot be omitted. They depend upon what we might call

heuristic, pragmatic, and contextual considerations. These are things like the

aims and goals of the scientist, what they consider acceptable accuracy, and

so on.

Similarly, the reasons for introducing falsehoods into one’s description of a

situation depend upon heuristic, pragmatic and contextual considerations. For

instance, establishing how to simplify a model or deciding what is a relevant

feature of a system that requires idealisation (Jones 2005, pg. 187). There is

a further question over idealisation, however: how is one able to gain useful

information from the use of false descriptions?22 In the case of isospin, the false

equating of the neutron and proton masses allowed an approximate symmetry

to be found. Isopsin, however, is now not recognised to be physically real. Any

attempt to explain an application of mathematics that involves abstraction or

idealisation will necessitate an account of scientific representation.

of how mathematics provides information about the world. i.e. I am not endorsing thatmathematics literally describes the world, or a structure of the world, or that it representsin a particular way, etc.

22This is a question that proves central to my later analysis of accounts of representation.See chapters 6 and 7.

28

Multiple Interpretations

Dirac’s prediction of the positron provides an example of where a single piece

of mathematics, the negative energy states, was given multiple interpretations.

They required some interpretation to either justify throwing them away (treat-

ing them as unphysical ‘waste products’ of obtaining a relativistic account of

how electrons behave) or to keep them. In classical mechanics, negative energy

solutions can be rejected as being non-physical and so are redundant. This can-

not be done in Quantum Mechanics, so some sort of physical interpretation

is required. Dirac originally interpreted them as the sea of negative energy

particles. However, only the ‘hole’ received any sort of physical interpretation.

We now understand the negative energy solutions as antiparticles, in this case

the positron.23

When introducing this example in §2.2.2, I posed the question “how can

one piece of mathematics have two very different interpretations yet both have

some predictive success?” This question can be rephrased within the context

of representation. It asks of an account of representation for an explanation

as to how we can interpret a single piece of mathematics in physically very

different ways, yet have predictive success with each interpretation. This falls

to an account of representation as interpretation of the representation vehicle

in terms of the target is a key part of an account of representation.

2.3.3 Novel Predictions

A feature of Dirac’s prediction of the positron that was not discussed above

is that the prediction was novel. That is, he was able to predict something

new that previous versions of Quantum Mechanics did not. In this case, the

positron and its behaviour. This problem has already been outlined as an23Steiner only focuses on Dirac’s prediction of the positron in the context of his argument

that prediction has undergone a change in definition, so has nothing to say on this particularproblem. Steiner’s arguments concerning prediction are discussed in the next section, §2.3.3.

29

intuitive worry concerning the prediction of the Ω− hadron via the use of

SU(3) and the prediction of the positron. Accounts of the applicability of

mathematics need to explain how these novel predictions can be legitimately

made. A challenge to the prediction of Ω− comes from Steiner, who argues that

it was achieved by ‘Pythagorean’ analogies, which are irrational for scientists

who advocate a naturalistic methodology. I will follow Bangu (2008) in arguing

that there is a difference between the prediction of the Ω− and the positron,24

outline Steiner’s objection to the way that Ω− was predicted, and then form

the problem of novel predictions in a similar way to Bangu.

The standard philosophical framework for novel predictions is provided

by Hempel’s (1962; 1965) Deductive-Nomological (D-N) account, which states

that predictions are derived from laws of nature and a set of initial conditions.

Bangu argues that this framework of prediction can account for “existential”

predictions (the prediction of some existing entity, as opposed to some physical

phenomena that occurs) quite easily (Bangu 2008, pg. 245). The prediction

of novel entities often arises as the result of some anomalous behaviour, as

an explanation of that behaviour. In the case of the positron, for example,

the negative energy solutions indicated some entity (first interpreted as the

vacant state in the electron sea) would behave as an electron with positive

charge. Once this behaviour was observed, further anomalous behaviour was

noted and then “explained away” by the positron (Bangu 2008, pg. 247). This

situation is different to the one surrounding the prediction of the Ω−. While

one can construct anomalies that require explanation,25 there is no mention of

interaction in the prediction of Ω−. The argument proceeds along apparently

Formalist lines that because the other spin-32 baryons behave in a way consis-

24French also highlights the difference by arguing that the D-N model is “too simplistic”to account for the prediction of the Ω− (French 1999, pg. 202).

25Bangu points to the nine baryons that already fitted the decuplet requiring some ex-planation. The finding of a tenth spin- 3

2baryon would allow for such an explanation, the

“law-like generalization H: ‘Spin- 32baryons fit the symmetry scheme” ’ (Bangu 2008, pg. 247).

30

tent with SU(3), there should be a tenth, the Ω−. The physicists “read off” the

characteristics of Ω− from the formalism, as opposed to having some empirical

evidence of them due to interactions they had participated in (Bangu 2008,

pg. 249).

This type of “reading off” prediction is described by Steiner as a ‘Pythagorean’

approach. He defines two such analogies (Steiner 1998, pg. 54):

Pythagorean: a mathematical analogy between physical laws “not para-

phrasable at t into non-mathematical language” - that is, an analogy

between two mathematically described laws.

Formalist: “one based on the syntax or even orthography of the lan-

guage or notation of physical theories, rather than what (if anything) it

expresses” - that is, by the mathematical formalism used, irrespective of

what it represents.

The reliance on Pythagorean analogies leads Steiner to argue that the meaning

of ‘prediction’ has in fact changed: “Prediction today, particularly in funda-

mental physics, refers to the assumption that a phenomenon which is math-

ematically possible exists in reality - or can be realized physically", that the

“concept of ‘prediction’ has itself become thoroughly Pythagoreanized” (Steiner

1998, pg. 161, 162). The idea is that the Ω− was predicted due to a Formalist

approach, where it is “read off” from the mathematical formalism. It is from

this understanding of prediction that Steiner argues against the rationality of

pursuing naturalistic approaches to science.

Bangu reconstructs Steiner’s argument and identifies an additional premise

in the prediction of the Ω−. He calls the additional premise the “Reification

Principle” (RP) and claims that this is the Pythagorean basis of the prediction

(Bangu 2008, pg. 248):

If Γ and Γ′ are elements of the mathematical formalism describing a

physical context, and Γ′ is formally similar to Γ, then, if Γ has a physical

31

referent, Γ′ has a physical referent as well.

The reconstruction focuses on whether the prediction of the Ω−, proceeding

via the RP as opposed to interactions, can be accommodated by a “physical”,

i.e. naturalistic, methodology (Bangu 2008, pg. 250). One way that one might

attempt to do so is to adopt a pluralism about methodology, where the RP is

incorporated as a novel part of the methodology. This is the approach Bangu

takes Steiner to be adopting. There is a difficulty in doing this, however, as the

RP is held to be counter to a naturalistic approach to science for the reason

that “there is no naturalistic explanation why RP could work, while the very

idea of a higher-level correspondence between mathematical objects and real-

ity reminds us of numerology, astrology, and other dubious practices involving

reifying mathematics” (Bangu 2008, pg. 250). Bangu’s characterisation of the

naturalist has them rejecting the RP as false, insisting that “there are count-

less situations in physics when the urged ‘principle’ just does not work. In

addition, no one can explain (without appealing to some additional mystical

verbiage) why such a principle works, even when it does.” Further suspicion is

cast on Steiner’s approach and redefining of prediction by noting his project to

argue against naturalism more generally (Bangu 2008, pg. 251). Given these

problems, it appears that one should reject the Pythagoreanisation of predic-

tion, at least in terms of “a systematic and all-encompassing methodological

move in (particle) physics”. However, doing so causes serious problems, as

the D-N model of prediction cannot accommodate (as argued above) the Ω−

prediction. Bangu finishes his argument as follows:

while the D-N model just does not work, the idea to go beyond it and

adjust scientific methodology along Pythagorean lines (by taking the

RP principle on board) is marred by serious conceptual problems. The

acceptance of RP as a credible element of physical methodology implies

not merely an innocuous re-definition of the concept of ‘prediction’, but

32

the abandonment of the very idea of a naturalistic viewpoint about the

world.

Bangu offers three possible responses to this worry, in an attempt to main-

tain a standard naturalist view point. These are (Bangu 2008, pg. 251-255):

i) The RP played a heuristic role in the prediction of the Ω−, by generating

a hypothesis rather than proving the existence of the Ω−;

ii) The RP played a justificatory role in the prediction of the Ω−;

iii) Though the Ω− was predicted due to the mathematics, it was done so

through interpreting the mathematics. It is this interpretation that is

significant.

Bangu rejects each of these in turn, then asks rhetorically whether the fail-

ure of these responses means that we should embrace the strict naturalistic

methodology and reject the practice of physicists who use the RP as being

unscientific. He holds that such a response would be unrealistic. He also refers

to French’s claims about the D-N model being too simplistic here,26 suggesting

that if this is the case, then we should reject it. Bangu then argues that the

real problem is not the predictive practice itself, but rather (Bangu 2008, pg.

255):

about understanding this practice. The problem presented here is thus

not a direct challenge for the working physicists, but for the physicist-

qua-methodologist; that is, a challenge for her ability to propose a sys-

tematic philosophical/methodological framework able to accommodate

this episode. Consequently, our problem is an instance of a broader kind

of concern: what is the most appropriate way to construe the relation

between philosophical/methodological standards and scientific practice.

26French (1999), pg. 202.

33

Bangu’s suggestion is to reject the strict naturalistic methodology in favour of

what he calls “methodological opportunism” (Bangu 2008, pg. 256):

By embracing this form of opportunism, the scientist can agree that

the D-N model cannot account for these predictions, and she can also

accept that the central role played by the mystical-sounding (RP) in

this reasoning is at odds with other naturalist convictions scientists of-

ten endorse. Yet she believes that this form of opportunism offers her

resources necessary to get over this tension; that is, she believes she is

entitled to retain the freedom to ignore this kind of conceptual deadlocks

and profit, opportunistically, from whatever might lead (even if unlikely)

to scientific progress.

Given the above arguments of Bangu, the novel prediction problem can be

seen to have several strands to it. First it asks for an explanation as to why

Formalist/Pythagorean prediction is successful (if one is available)? (This

could be understood as asking for whether an explanation of why the RP

succeeds is available. Bangu’s naturalist claims that such an explanation is not

available.) Second, it asks whether we can provide a consistent methodological

understanding of how novel predictions are made? (Whether we can have a

consistent methodological treatment of the positron and Ω− predictions, of D-N

type and Formalist/Pythagorean type predictions.) Third, if we can have such

a consistent methodological treatment, can it be one that conforms with the

strict naturalistic view? Or must it be Bangu’s methodological opportunism, or

similar? Fourth, if we cannot have such a consistent methodological treatment,

must we accept Steiner’s pluralism, or similar?

34

2.3.4 Isolated Mathematics and Steiner’s

Epistemological Problem

The Pythagorean analogies introduced in the section above play a key role in

Steiner’s epistemological problem. He takes this problem to be the “unrea-

sonable effectiveness of mathematics in the natural sciences” highlighted by

Wigner (1960). In discussing the success of SU(3) in predicting the quarks,

given the use of Pythagorean analogies, Steiner states that:

Mathematicians, not physicists, developed the SU(3) concept, for rea-

sons unconnected to particle physics. They were attempting to classify

continuous groups, for their own sake.

Because the SU(3) story is not isolated, there are physicists who main-

tain that mathematical concepts as a group, considering their origin, are

appropriate in physics far beyond expectation. . . . It concerns the appli-

cability of mathematics as such, not of this or that concept. . . . It is the

question raised by Eugene Wigner about the ‘unreasonable effectiveness

of mathematics in the natural sciences’ Wigner (1960).

He then draws the following argument from Wigner (Steiner 1998, pg. 47):

(1) Mathematical concepts arise from the aesthetic impulse in humans.

(2) It is unreasonable to expect that what arises from the aesthetic

impulse in humans should be significantly effective in physics.

(3) Nevertheless, a significant number of these concepts are signifi-

cantly effective in physics.

(4) Hence, mathematical concepts are unreasonably effective in physics.

There are two threads the problem that I wish to draw out. First is the

issue of what I call ‘isolated’ mathematics - mathematics that originated in

isolation from considerations of applications. Steiner claims that SU(3) is a

35

piece of isolated mathematics. Second, the main force of Steiner’s epistemolog-

ical problem, and the anti-naturalist conclusions he draws from it, is the claim

that what decides whether something is ‘mathematical’ or not - the mathemat-

ical criteria - is based on human aesthetic (and tractability) considerations. I

will argue below that isolated mathematics, the first thread, is a problem for

accounts of applicability. First I will survey Bangu’s arguments that the prob-

lem of mathematical criteria, the second thread, is not an issue that needs to

be answered by accounts of applicability, but by philosophers of mathematics.

Consequently I will restrict the epistemological problem, and hence rename it,

to the problem of isolated mathematics.

Steiner argues that the use of Pythagorean analogies should lead to a re-

jection of naturalism. To explain the apparent ‘mystery’ of mathematics being

applicable at all, he concludes that there is some special connection between

the human mind and the world. Bangu argues that Steiner’s conclusion (cos-

mological anthropocentrism) rests upon two conditions (Bangu 2006, pg. 36):

First, if there is a record of systematic employment of mathematically

guided successful strategies in the invention of new theories, and no non-

mathematical strategy is available. Second, if these successful, mathe-

matically guided, strategies are indeed anthropocentric, but this holds

only if mathematics is an anthropocentric concept.

Where mathematical anthropocentrism is defined as follows (Bangu 2006, pg.

35):

Mathematical anthropocentrism is a view arrived at by examining how

mathematics develops as a discipline and not by analyzing specific math-

ematical concepts applied in physics. In this context, ‘anthropocentric’

is a predicate that applies not to particular mathematical concepts, but

to the nature of the criteria employed when concepts are added to the

corpus of the discipline.

36

Bangu provides two examples27 that show it is not the case in the history of

mathematics that the mathematical criteria are exclusively anthropocentric,28

thus Steiner’s argument is unsound. I agree with Bangu that the problem of

mathematica criteria is a problem for philosophers of mathematics; it does not

fall under the umbrella of problems of the applicability of mathematics. As I

am following the applications first approach, issues concerning what is or isn’t

mathematics are of no relevance to answers of the problems of the applicability

of mathematics.

Isolated Mathematics

The history of how mathematical concepts are discovered/created does not

depend upon one’s metaphysics. It is the fact that the mathematics originated

for purely mathematical reasons, as opposed to being discovered or created for

a specific physical situation that causes an apparent mystery. The reason that

isolated mathematics is claimed to be problematic is that there appears to be

no reason why it should allow any information to be found about the world

on previous accounts of applicability. Compare such a piece of mathematics

with the development of the calculus. This was developed in response to four

problems according to Kline: establishing the velocity and acceleration of a

particle at an instance; finding the tangent of a curve (and even establishing

a global definition of tangent); finding the maximum and minimum value of

functions; and the length of curves (Kline 1972, pg. 342-343). Solutions were

found independently to these problems and some links had even been found

between the solutions before the calculus, but the solutions lacked the general-

ity provided by the work of Newton and Leibniz (Kline 1972, pg. 356). Given

27The debate between Euler and d’Alembert over what a function is (Bangu 2006, pg.37-38), and over the Axiom of Choice (Bangu 2006, pg. 38-39).

28Some instances of establishing whether a concept is mathematical are explicitly an-thropocentric. He quotes Zermelo, who “insists that the [Axiom of Choice] is ‘necessary forscience’ (Zermelo 1967, pg. 187)” (Bangu 2006, pg. 40).

37

the history of the calculus, one can see the difference between isolated mathe-

matics and applied mathematics is that the applied mathematics is explicitly

discovered or constructed in connection with a problem and so is formulated

in a way conducive to being applied.

Whether a mathematical concept falls under the banner of ‘isolated math-

ematics’ is a matter of historical fact. Steiner claims that SU(3) is a piece

of isolated mathematics, but it is not as clear a case as Steiner presents it as

being. There is evidence that group theory in general was developed before

any applications: Maddy (2007) argues this, heavily relying on Kline (1972).

However, Wigner adopted group theory to work on symmetry in crystals and

French (2000) argues that the application of SU(3) to the light hadrons was

not a result of isolated mathematics. This is due to the work of Weyl and

Wigner in the early 20th century. They further developed group theory so that

it could be applied to both the fundamental symmetry principles of Quantum

Mechanics and various problems to provide simple (or even new) methods of

calculation. There are other cases of isolated mathematics, which are of a

less controversial nature. For example, the mathematics used to establish the

tightest packing possible for spheres has been used to model modem signals

and their associated noise as eight dimensional spheres.29

The problem of isolated mathematics is the requirement for an explanation

of how mathematics that originated in isolation from any considerations of

application can nevertheless be found to be applicable to physical situations.

This concludes the discussion of Steiner’s problems that I will answer in this

thesis. I will now turn to the semantic and mathematical explanation prob-

lems, and explain why they will not be answered. The semantic problem asks

29This is the most clear cut cases of isolated mathematics from seven examples found inRowlett (2011). Some of the examples given by Rowlett come close to a related problemdescribed by Maddy as one of ‘transfer’ (Maddy 2007, pg. 333). A solution to the isolatedmathematics problem should also solves this ‘transfer’ problem.

38

for a consistent logical interpretation of statements that involve mathematics.

The mathematical explanation problem asks whether mathematics contributes

explanatorily to science, and if so, how it is capable of doing so.

2.3.5 The Semantic Problem

Accounts of applied mathematics have to make sense of arguments that involve

both statements of pure mathematics and ‘mixed statements’ - statements

that have both mathematical and physical terms in them. One such mixed

statement is ‘the satellite has a mass of 100kg’. This statement might occur

in an argument over the force required to launch the satellite into orbit. The

semantic problem can be immediately recognised from such arguments. The

problem is the requirement for “a constant interpretation for all contexts -

mixed and pure - in which numerical vocabulary appears” (Steiner 1998, pg.

16). The problem arises as in pure mathematical statements numerical terms

appear to be singular terms that refer to numbers, while in mixed statements

they appear to be predicates (in this case, about the mass of the satellite).

Steiner’s solution to the semantic problem is to follow Frege in taking all

numerical terms to be singular terms (Steiner 1998, pg. 16-17).

Pincock holds that the mixed statement ‘the satellite has a mass of 100kg’

should be understood as follows: “the statement seems to be about the satellite,

the real number 100, the standard gram (assuming that there is such a thing),

and some complicated relation between them” (Pincock 2004a, pg. 139). While

it seems that Pincock wishes to follow Steiner in taking all numerical terms to

be singular terms, Pincock argues that the truth of mixed statements depends

on the relation between the world and the mathematical domain, rather than

the existence of numbers (Pincock 2004a, pg. 145). This implies that the

interpretation of the numerical term is secondary to the relation between the

physical world and the mathematical domain.

39

In his (2012) Pincock explicitly shies away from attempting to answer the

semantic problem, due to difficulties related to interpreting natural language

sentences and their relationship with scientific representation (Pincock 2012,

pg. 171-172). This is partly because the semantic problem goes further than

questions of how to interpret numerical terms. Applied mathematical argu-

ments also include functions, algebra and other mathematical terms in their

mixed statements. Establishing how to interpret all mathematical terms in a

consistent way goes beyond what Pincock wishes to achieve in his (2012), and

goes beyond what I intend to argue for in this thesis. I will therefore say no

more on the semantic problem.

2.3.6 The Mathematical Explanation Problem

I have left the problem of mathematical explanation to last for two reasons.

First, there is no single philosophical theory for all cases of explanation. Sec-

ond, it is not settled whether mathematics can explain physical phenomena.

This immediately leaves one unclear on how to frame the problem, and com-

plicates what form an answer might take. I will briefly outline the key ideas

and players in the debate over this issue, mostly by following Lange (2013).

Due to the amount of work required to settle the two debates, however, I will

not attempt to answer this problem in this thesis.

The typical starting point for a discussion concerning scientific explanation

is the Deductive-Nomological (D-N) model of Hempel.30 Here explanation

is understood to be a deductive argument, where the explanandum-event is

deduced (and therefore explained) from general laws and particular facts (the

premises of the argument being the explanans). A D-N explanation has the

following, syllogistic form (Hempel 1962, pg. 686):

30See Hempel (1962), Hempel & Oppenheim (1948)

40

C1,C2, . . . ,Ck

L1, L2, . . . , Lr

E

Unfortunately, the D-N model counts various non-explanations as explanatory,

e.g. a flag pole’s shadow counting as an explanation of its height. It has been

argued that the solution to this problem is to include causal relations in one’s

explanations. Lange points to several philosophers who hold the stronger view

that all scientific explanations are causal explanations (Lange 2013, pg. 486-

487).31

This view faces serious difficulty with accounting for mathematical expla-

nations. It is commonly held that mathematical explanations, whatever they

might consist in, make some appeal to a mathematical fact, feature or result.32

The key consequence of this is that mathematical explanations are held to be

non-causal. Any account of scientific explanation that is exclusively causal

will thus rule out mathematical explanations in principle. Other possible ac-

counts of explanation that might be capable of accommodating mathematical

explanation include the unification account,33 and abstract or structural ex-

planations34

The debate concerning whether mathematics can explain physical phenom-

31Lange quotes Salmon (1977, 1984), Lewis (1986) and Sober (1984) amongst others asbeing in support of this claim. He also points to the more recent accounts of Strevens (2008)and Woodward & Hitchcock (2003) due to their emphasis on causal connections.

32Baker’s (2005) appeal to the number theoretic theorem that prime periods minimiseintersection compared to non-prime periods is an example of this. Lange’s particular con-ception of mathematical explanations rejects this, arguing that distinctive mathematicalexplanations appeal to facts that are “modally stronger than ordinary causal laws” (Lange2013, pg. 491). This disagreement between Baker and Lange is relevant to the idea thatthere is no clear idea of what constitutes a mathematical explanation, rather than the pointI am making here.

33Kitcher (1981, 1989)34Pincock’s (2007) abstract explanations are called structural explanations by McMullin

(1978). In his (2012), Pincock refers to the explanation of the bridges of Königsberg asmathematical explanation, due to being an abstract (acausal) representation. Batterman(2001, 2010) can also be understood as offering up a structural account of explanation.

41

ena has proceeded in a series of papers focusing on the indispensability argu-

ment.35 The idea at the heart of the debate is that for mathematics to be

indispensable such that we can conclude that numbers exist, it has to be in-

dispensable in the right way, namely, explanatorily indispensable. The debate

then proceeds by the identification of putative mathematical explanations of

physical phenomena, such as the prime number life cycles of periodic cicada

and the hexagonal shape of honeycombs. Arguments are then put forward in

an attempt to determine whether the explanation really does depend on some

piece of mathematics, a mathematical fact, etc. Recent work, such as Lange’s,

has attempted to gain a firmer grip on the notion of mathematical explana-

tion, and identify what the putative explanations actually depend on to do

their explanatory work (e.g. causal facts, laws, or in Lange’s case, modally

stronger facts).

In this section I have distinguished nine problems that fall under the banner

of the problems of the applicability of mathematics. I’ve argued that three of

these problems do not need to be answered in this thesis, and that six need

to be. I will now answer the applied metaphysical problem and explain why

the answer to this problem should serve as a basis for an answer to the other

problems I will be answering.

2.4 Solving the Applied Metaphysical Problem

I have argued above that previous attempts to solve the problems of the appli-

cability of mathematics have approached the issues from the wrong direction.

Applications of mathematics by physicists do not generally involve consider-

ations of ontology. These questions are secondary to the application; they

35These papers include: Melia (2000, 2002), Colyvan (2002), Baker (2005, 2009) andSaatsi (2007, 2011).

42

ask what the mathematical statements refer to, rather than how the appli-

cations work. I have argued for the problems above to be framed in ways

which make as few assumptions about the ontology of mathematics as pos-

sible. Any solutions to these problems should be equally minimal in their

ontological commitments, preferably making no commitments at all. The first

step to solving these problems is to answer the applied metaphysical problem.

An ideal answer should be one which forms the basis for an account of math-

ematical scientific representation, which is capable of solving the problems of

representation and hopefully be extended to solve the other (semantic, math-

ematical explanation) problems. I believe that Structural Relation Accounts

(SRAs) of applicability can do this. These accounts do not commit one to be

a realist or anti-realist about mathematics, as, in general, these accounts are

compatible with numerous realist and anti-realist positions.

Before arguing for the SRAs, I wish to make note of Pincock’s analysis

of Steiner’s answer to the applied metaphysical problem. As Steiner endorses

a Fregean ontology of mathematics, his answer to the applied metaphysical

problem for counting is a one-to-one correspondence relation. This is extended

to the rest of mathematics via adopting a set-theoretic approach where we can

have impure sets, i.e. sets that contain physical objects as individuals (Steiner

1998, pg. 28). Pincock is rather unsatisfied with this as an answer to the

applied metaphysical problem (Pincock 2012, pg. 173, my emphasis):

If this set-theoretic solution is deemed adequate, then it shows how low

the bar is set to solve the [applied] metaphysical problem. Steiner raises

the issue by alluding to a gap between mathematics and the physical

world. A bridge across this gap need only show that mathematics is

related in some way to the physical world. But as Steiner’s own set-

theoretic response indicates, there is no requirement that the bridge do

anything else. It need not illuminate what, as [Pincock has] continually

put it, mathematics contributes to the success of science.

43

I agree with Pincock’s analysis here: an answer to the applied metaphysical

problem is straightforward. All it requires is a relation be provided between

the world and a mathematical domain. The requirement that the “bridge”

do anything else, i.e. restrictions on what type of relation might adequately

relate the world and the mathematical domain, comes from the other problems

of applicability. This is why the only restriction I place on an answer to the

applied metaphysical problem be that it is capable of being extended to answer

the other problems.

SRAs hold that the applicability of mathematics is due to structural rela-

tions between physical structures and mathematical structures. These struc-

tural relations are structure preserving: they take the structure of physical

entities or properties to areas of mathematics that display a given level of

structural similarity.36 For example, when numbers are applied to state the

length of some object, a structural relation will be involved. This relation will

have to take the structure of the length property of the object to the relevant

part of the real number structure. It will also have to include information

on the length property in terms of its unit. e.g. “that the mapping takes all

meter sticks to 1, all objects that are half as long as a meter stick to 12 , etc.”

(Pincock 2004a, pg. 147). There are various types of structural relations that

may fulfil the role required by applications of mathematics; isomorphisms and

homomorphisms are the most commonly appealed to structural relations.37

Which structural relations are taken to constitute the representation relation

36The simple explanation given here is naïve. As rightly pointed out by Bueno andColyvan, the parts of the world involved in our scientific theories are not always actualconstituents of the world. They adopt a standard understanding of a structure as “setof objects (or nodes and positions) and a set of relations on these)” and remind us that“the world does not come equipped with a set of objects (or nodes or positions) and setsof relations on those. [(The view of structure above originates from Resnik (1997) andShapiro (1997).)] These are either constructions of our theories of the world or identified byour theories of the world” (Bueno & Colyvan 2011, pg. 347). How theories are capable ofcarving up the world in successful ways will fall under an account of representation and sowill be addressed in the next chapter.

37Loosely, an isomorphism maps all of the structure from the domain, D, to the codomain,D′. A homomorphism maps some of the structure of D to D′.

44

is one of the main distinguishing features of the different SRAs.

The structural relations involved in these accounts are taken to be external

relations. These are relations that do not “involve the criteria of identity of

the mathematical objects” (Pincock 2004a, pg. 145). Admitting only exter-

nal relations in applications of mathematics has the advantage of limiting the

ontological commitments one is required to make in understanding mathemat-

ical applications. An internal relation is defined thus: “a stands in an internal

relation R to b just in case aRb’s obtaining is involved in a’s criteria of iden-

tity” (Pincock 2004a, pg. 140). Pincock argues that using internal relations

in the hypothetical reasoning involved in science forces one to adopt modal

realism (Pincock 2004a, pg. 143-144). Given that I am attempting to make as

few ontological commitments about mathematics as possible I cannot justify

making commitments about modality. Field’s approach is described by Pin-

cock as a ‘no relation’ account. As Field denies that there are mathematical

entities there can be no relations between the physical world and mathemat-

ical entities. Pincock argues directly against Field’s account, rather than no

relation accounts in general. This begs the question as to whether no relation

accounts are in principle compatible with structural relation accounts. Bueno

and Colyvan hold this view, as they claim that Field would be able to make

use of the Inferential Conception (Bueno & Colyvan 2011, pg. 368). I will

not settle this debate here. I take it to be part of the work required to answer

the pure metaphysical question to establish which mathematical ontologies are

consistent with the solutions to the problems I argue for throughout this the-

sis. However I will commit to the claim that the structural relations are be

external relations, as this maintains my goal of making minimal ontological

commitments.

SRAs answer the applied metaphysical question as follows: the mathemat-

ical domain is related to the physical domain in virtue of a structural relation.

45

The world can be understood as conforming to a particular structure which

can be found in the mathematical domain. Any piece of applied mathematics

is applicable in virtue of this sharing of structure, via the structural relation.

This is a very straightforward, and general answer. Notice that it does not

explain how the structural relation is capable of identifying the structures, nor

how any particular piece of mathematics is applicable to any particular piece of

the world. The answer to these questions needs to be provided by an account

of representation, as questions over a particular piece of the world having a

particular piece of mathematics applied to it are actually questions over how

a particular piece of mathematics is capable of representing that piece of the

world.

2.5 Conclusion

In this chapter I have argued for nine individual problems that fall under

the applicability of mathematics. I have employed structural relations to an-

swer the applied metaphysical problem, with the promise that accounts that

make use of such relations can be developed that can answer the problems of

representation, the novel predictions problem and the isolated mathematics

problem. Specifically, these accounts need to be developed into accounts of

representation. But what are the requirements for a successful account of rep-

resentation? Are these requirements the same for all forms of representation,

or is there something unique about scientific representation? Can structural

relations satisfy these conditions? These are some of the questions I introduce

and begin to answer in the next chapter.

46

3 | Representation

3.1 Introduction

Representation is a very wide ranging concept. It appears not only in science,

but also in art, psychology, cognitive science and linguistics. As such there

are a huge number of issues involved with its philosophical analysis. The first,

and perhaps most serious, of these issues is whether there is a fundamental

representation relation, consistent across all types of representation. Possible

answers to this raise further questions: if there is a fundamental relation, what

distinguishes the various types of representation, if they can be distinguished

at all? If there is not a fundamental relation, then how are we to understand

the various activities that fall under the term ‘representation’?

In this chapter I will highlight only a few of these issues, those that I

take to be most pressing on scientific, and mathematical scientific, represen-

tation. I will follow the lines of argument set out in the literature in this

regard. I will begin my investigation by focusing on the question of whether

there is a “special problem of scientific representation”, a question posed by

Callender & Cohen (2006). By concentrating on this question I will be able

to explore whether there is something unique to scientific representation that

distinguishes it from putatively distinct types of representation, such as aes-

thetic representation, or whether there is a unified notion of representation. I

will argue in line with Contessa (2007; 2011) that one can identify epistemic

47

representation as a distinct form of representation, rejecting a unified notion

of representation. One can establish this distinction by noting that one uses

epistemic representation to gain information about a representational target

(and so the presence of surrogative inferences are a distinguishing feature of

epistemic representation), whereas aesthetic representation is used to convey

aesthetic values (e.g. beauty, emotions, etc.).

In following Contessa’s arguments, I reach a conclusion that is in part in

agreement with Callender and Cohen: there is nothing special about scientific

representation. However, it is in part in disagreement with Callender and Co-

hen, as I reject a uniform notion of representation; there might not be anything

special about scientific representation, but this is compared to other instances

of epistemic representation. Scientific representation is simply epistemic rep-

resentation undertaken by scientists or where the target is the world. There is

still a distinction between epistemic and other forms of representation. Con-

tessa’s arguments in favour of epistemic representation do not complete his

project, however. He identifies the need for an account of faithful epistemic

representation: representation where at least one of the surrogative inferences

from the representational vehicle to the target is sound. I will argue that, in

general, structural relation accounts of representation are the accounts that are

best placed to provide such accounts. Outlining and explaining how Pincock’s

Mapping Account ((2012)) and the Inferential Conception (Bueno & Colyvan

(2011), Bueno & French (2012)) can be understood as accounts of faithful

epistemic representation will be started in the next chapter. Evaluating the

ability of these two accounts to sufficiently explicate the relationship between

how faithful a representational vehicle is and its usefulness constitutes the rest

of the thesis.

48

3.2 What is Representation?

A first pass at analysing representation is “X represents Y ”. This, however,

is rather uninformative. We would like to know how X represents Y . In

virtue of what does X represent Y ? The first pass ignores a crucial element of

representation, that it involves agents. It is not so much that “X represents

Y ”, but that an agent uses X to represent Y . This is noted in the literature in

a couple of slogans. van Fraassen points to the analogous example of Mead’s1

reflections on teacups: “If there were no people there would be no teacups,

even if there were teacup-shaped objects. For ‘there are teacups’ implies that

‘there are things used to drink tea from’ which in turn implies ‘there are tea-

drinkers’.” (van Fraassen 2010, pg. 25). Giere makes use of the slogan “no

representation without representers” (Giere 2006, pg. 64, endnote 13). The

idea behind these slogans is that representation is an intentional activity: an

agent uses a entity to represent a target. This notion of use needs spelling out.

van Fraassen argues that ‘use’ encompasses many contextual factors, which

prompts us to look at the practice of representation to properly analyse ‘repre-

sentation’ (van Fraassen 2010, pg. 23). He argues that in some cases distortion

is crucial to the success of representation, yet in others is a cause of misrep-

resentation (van Fraassen 2010, pg. 13-14). The important conclusion from

this discussion is that representation is representation as. Our representations

involve using a vehicle to represent the target as something, where that some-

thing is contextually dependent upon our use, the aims we are seeking to fulfil

in adopting the representation. We can therefore expand our initial analysis of

representation to: “A uses X to represent Y as F ”;2 or “S uses X to represent

1van Fraassen references McCarthy (1984) for this observation.2van Fraassen initially presents a minimum schema of “X represents Y as F ” and argues

that this is the minimum form the schema takes (van Fraassen 2010, pg. 20). His idea isthat it gets modified according to the use at hand.

49

W for purpose P ”.3

I will argue below for the view that representation is an activity, where

the success of this activity is dictated by the purpose involved. That is, the

purpose to which someone adopts a representation dictates how successful that

representation is. ‘Purpose’ and ‘use’ are cashed out contextually. I will make

use of this to argue that the type of representation fills in some of the ‘purpose’

and ‘use’ criteria, and that this dictates what sort of representation relation

should be adopted for particular types of representation.

While agents are necessarily involved in representation, the above schema

do not make it clear how an agent should be involved in the representation

relation. Should they be related via the representation relation to both vehicle

and target? Should they be related by another relation to the only the vehicle,

or only the target? The answer to this question varies from account to account,

and so will be provided in the next chapter rather than here.

3.3 A Special Problem for Scientific

Representation?

Whether the above schema of “A uses X to represent Y as F ” should be

expanded or refined, and how, will depend on the answer to two questions.

First, are there any special features or purposes of representation in general?

Second, is there is some special feature or purpose of scientific representation

specifically, that sets it apart from other forms of representation? Fortunately,

attempting to answer the second of these questions will provide an answer to

the first.

The title of this subsection comes from the title of a paper by Callender

and Cohen (CC), who pose the problem specifically for scientific models. They

3Giere argues for this formulation. S is an individual or group of scientists, or a scientificcommunity, while W is an “aspect of the real world” (Giere 2006, pg. 60).

50

ask how can models be about the world given that they can have features the

world lacks, or lack features the world has. This is further complicated if it is

right that models are not truth-apt (or approximately-truth-apt). Their paper

distinguishes four questions that they take to be addressed by accounts of

scientific representation, though they do not think that any particular account

has addressed all four questions. These questions are (Callender & Cohen

2006, pg. 68-69):

(CQ) Constitution Question: What constitutes the representational re-

lation between a model and the world?

(DQ) Demarcation Question: What is distinct about scientific represen-

tation?

(NI) Normative Issue: What is it for a representation to be correct?

(EQ) Explanatory Question: What makes some models explanatory?

CC introduce the NI in response to what Morrison calls the “heart of the

problem of representation”, namely the question “in virtue of what do models

represent and how do we identify what constitutes a correct representation?”

(Morrison 2008, pg. 70). They argue that there are two problems in this quo-

tation: the ‘in virtue of’ question being the constitution question; and the

‘correct representation’ question broaching the normative issue. They further

argue that proponents of the ‘models as mediators’ view4 push too strongly on

the idea that the “representational and explanatory capacities of a model are

interconnected”.5 They agree with this claim if one understands the ‘intercon-

nection’ to be due to the NI and EQ presupposing answers to the constitutive

ones, the CQ and DQ, but no more than that (Callender & Cohen 2006, pg.

69).4See Morrison (1999).5Callender and Cohen point the reader to page 40 of Morrison’s (1999) in support of

this claim.

51

CC favour what I consider to be a unified account of representation. They

argue that “the varied representational vehicles used in scientific settings . . .

represent their targets . . . by virtue of the mental states of their makers/users”

(Callender & Cohen 2006, pg. 75). All representation is to be reduced to mental

representation. Thus there is no special problem of scientific representation

as it is merely reduced to mental representation where the agents doing the

representation are “scientists and their audiences are either fellow scientists or

the world at large” (Callender & Cohen 2006, pg. 83).

Due to their adoption of this unified account of representation, CC argue

that issues related to the misrepresentation involved in scientific models are

not as important as the literature makes them out to be. They do this by

appealing to examples from other types of representation (Callender & Cohen

2006, pg. 70). For example, they rhetorically ask whether linguists are con-

cerned that “the marks ‘cat’ aren’t furry or that cats lack constituents that

are parts of an alphabet” in order to make the point that such questions are

“bad”. If one is making use of a unified account of representation, they argue,

these questions should also be bad questions to ask of scientific representation.

Their argument against taking misrepresentation due to abstraction, idealisa-

tion, and so on, seriously is that all forms of representation have something in

common that make such questions inappropriate (perhaps even due to cate-

gory mistakes - representations are not the sort of thing one should ask such

questions of). Obviously a rejection of their unified account of representation,

and the adoption of a non-unified notion of representation, leads to these issues

having greater significance than CC take them to have, in accordance with the

other literature on scientific representation. These issues are most relevant to

the NI, though also play a role in establishing an adequate answer to the CQ

(as the representation relation has to be able to account for misrepresentation

in a satisfactory manner).

52

I agree with CC’s position on the relationship between the questions: an-

swers to the NI and EQ will depend upon the answers provided for the CQ and

the DQ. I will therefore first provide answers to the CQ and DQ in this chapter,

arguing that the answer to the CQ depends upon an answer to the DQ and

against a unified account of representation in the process of this. Although I

agree with CC’s conclusion that there is nothing special about scientific rep-

resentation, this is for different reasons, namely that scientific representation

is epistemic representation performed by scientists. Thus the more interest-

ing DQ is not what counts as scientific representation, but what counts as

epistemic representation. The answer to the CQ then must be capable of pro-

viding epistemic representation. I will, in the later chapters, argue that the

Inferential Conception and Pincock’s Mapping Account are plausible accounts

of faithful epistemic representation. An answer to the NI will be given in the

process of establishing which, if either, of the accounts succeeds as an account

of faithful epistemic representation. I will not address the EQ. In the context

of mathematical scientific representation, I see this question to be asking the

same as the mathematical explanation question of the previous chapter and so

will leave it for the same reasons provided there.6

3.3.1 The Constitution Question

I take the Constitution Question to be the most important question to answer

if one is attempting to establish a unified account of representation across all

forms of representation. If one is able to show that all types of representation

occur due to one type of relation then one has gone most of the way to provid-

ing a unified account of representation. Thus, if I wish to show that a unified

account of representation is not possible, I merely have to find an example of

representation that must proceed via a different type of relation to scientific

6See §2.3.6.

53

representation. One way of doing this is to pay attention to the purpose rep-

resentations are put to. If a particular type of representation has a distinct

purpose, this purpose supplies the answer to the DQ. Further, in order for the

representation to be able to fulfil that purpose, the representation relation for

different types of representation may have to be constituted by a different type

of relation. That is, an answer to the CQ will depend on an answer to the

DQ. The answer to the DQ will establish certain requirements that the repre-

sentation has to fulfil to be a representation of that type. The representation

relation must be capable of allowing the vehicle to fulfil these requirements.

Thus if I am able to identify a unique feature of scientific representation, I

hold that a unified account of representation is not possible.

My argument that scientific representation has a unique feature will proceed

by first surveying arguments put forward against the similarity and structural

relation accounts by Suárez and Frigg. I will begin with these arguments

due to the conclusion of the previous chapter, that accounts of representation

based on structural relations could be developed to answer the problems of

representation, novel predictions and isolated mathematics. I will find these

arguments to be successful against similarity accounts, but unsuccessful against

structural relation accounts. This survey will also indicate that inferences are

important to scientific representation. This will prompt me to switch focus to

the DQ, in order to establish the role inferences play in scientific representation,

namely whether they are a distinguishing feature of scientific representation.

There are three broad categories of accounts of scientific representation

which supply different answers to the CQ: similarity; structural relation; and

the Inferential Conception Approach. Giere (1988, 2004) argues for the rep-

resentation relation to be one of “similarity” or “fit” between a model and the

world. The structural relation approach includes those who propose the rela-

54

tion is some form of partial-morphism7, Pincock who takes there to be other

structural relations involved8 and various other authors who pick a particular

form of morphism.9 Finally, there is Suárez’ Inferential Conception Approach10

which involves refraining from specifying any particular type of relation, but

rather explicating conditions a particular type of relation would have to meet

in under to be a representational relation.

In the literature the similarity and structural relation accounts are often

held to fail for the same reasons. Suárez (1999, 2003) and Frigg (2002, 2006)

provide the same sorts of arguments along these lines. In Suárez’ case, these are

clearly meant to show that superiority of his Inferential Conception Approach.

I will follow the arguments of Suárez (2003) here.11 Five arguments against

the structural relation and similarity accounts are put forward:

The Argument From Variety: similarity & isomorphism do not ap-

ply to all representational devices;

The Logical Argument: similarity & isomorphism do not posses the

logical properties of representation;7Such as French (2003), Bueno & Colyvan (2011), Bueno & French (2011, 2012)8Pincock (2012).9For example Bartels (2006), who argues for using homomorphic mappings. I will not

discuss Bartels’ account, as I take it to collapse into a partial structures account and thereforefavour the Inferential Conception. The major problem with Bartels’ account is that thenotion of homomorphism he employes is too rigid to be able to accommodate all casesof misrepresentation. Bartels is aware of this, and in attempting to fully accommodatemisrepresentation he argues for a weakening in the conditions that hold for a homomorphismis occur (Bartels 2006, pg. 10). He then points to the partial structures framework as anexample of a structural account which has adopted this kind of weakening (Bartels 2006, pg.10, footnote 3). As Bartels’ account is not developed further along the lines of the partialstructures framework, it should be rejected in favour of the account that is, the InferentialConception.

10Suárez (2003).11Suárez offers his arguments against a specific version of the structural relation accounts,

namely a simple isomorphism account: “A represents B if and only if the structure exempli-fied by A is isomorphic to the structure exemplified by B” (Suárez 2003, pg. 227). He alsooffers them against a simple similarity account: “A represents B if and only if A is similarto B”. He does go on to argue that these arguments can be extended to the “weaker” and“amended” versions of the isomorphism account (Suárez 2003, §5-6). I respond briefly toSuárez’ claims in §3.3.1 - Further Arguments, and Amended & Weakened Versions. Theseweaker and amended versions are still less sophisticated than the accounts I will endorse inthe next chapter.

55

The Argument From Misrepresentation: similarity & isomorphism

do not make room for the ubiquitous phenomena of mistargeting and/or

inaccuracy;

The Non-necessity Argument: similarity & isomorphism are not nec-

essary for representation - the relation of representation may obtain even

if similarity & isomorphism fail.

The Non-sufficiency Argument: similarity & isomorphism are not

sufficient for representation - the relation of representation may fail to

obtain even if similarity & isomorphism hold.

The argument from variety can be avoided by proponents of the similarity

and structural relation accounts in a legitimate way, according to Suárez, by

leveraging a distinction he draws between the means of representation, and

the constituents (Suárez 2003, pg. 230):

Means of representation: At any time, the relation R between A and

B is the means of representation of B by A if and only if, at that time, R

is actively considered in an inquiry into the properties of B by reasoning

about A.

Constituents of representation: The relation R between A and B

is the constituents of the representation of B by A if and only if R’s

obtaining is necessary and sufficient for A to represent B.

What Suárez is identifying by the ‘means of representation’ is the relation be-

tween A and B that is used in the drawing of a particular surrogative inference

about the target (Suárez 2003, pg. 229).12 There are many relations that exist

between a representational vehicle and its target; one of these, say an isomor-

phism, might be useful for obtaining one set of surrogative inferences, while12Suárez follows Swoyer (1991) in taking “the main purpose of representation [to be]

surrogative reasoning”. I will assume that this is the case while discussing Suárez’ arguments,and will argue in favour of this being the case for scientific representations in §3.4.4.

56

another, say similarity (the sharing of properties), might be useful for obtain-

ing some other set of inferences. While addressing the CQ, only arguments

aimed at showing that similarity and isomorphism fail to be the constituents

of representation need to be answered. The means of representation is rele-

vant to the DQ; if structural relations cannot provide the means of scientific

representation, then they will not satisfy any particular conditions of scientific

representation that distinguish it from other forms of representation.

I will briefly outline these arguments below, and respond to them in ac-

cordance with the structural relation accounts in general (where possible). I

take these arguments to be damaging to the similarity account of represen-

tation.13 I will not outline the argument from variety due to my response to

the DQ. Below, I will argue that the answer to the DQ sets limits on the CQ,

specifically that the relation that provides the answer to the CQ has to be

capable of providing surrogative inferences. Thus I reject Suárez’ distinction

that one can draw surrogative inferences from a different relation to the one

that constitutes the representation relation. According to my arguments, for

structural relations to constitute the representation relation, they have to be

capable of providing the surrogative inferences in all cases of scientific represen-

tation, hence the distinction collapses, and the variety argument is implicitly

answered.

The Logical Argument

A representation relation is held to have a particular set of logical properties,

specifically that it is asymmetric, non-reflexive and non-transitive. Similar-

ity is reflexive and symmetric, while isomorphism is reflexive, symmetric and

13I do not mean to deny that one is capable of responding to these arguments fromthe position of a similarity account. Rather, due to the focus of this thesis, I take thesearguments and the responses from the structural relation accounts to be sufficient for meto adopt the structural accounts from this point on. As a result, I will not provide anyresponses in favour of similarity accounts.

57

transitive.

Structural relation accounts have various options open to them. Suárez’

argument is aimed at isomorphisms, so any structural relation account that can

make use of either partial isomorphisms or other morphisms than isomorphisms

(such as homomorphisms) should be able to provide the asymmetric and non-

transitive properties. The non-reflexive property is more challenging to provide

for the partial isomorphism account, as partial structures can be partially

isomorphic to themselves. However, if one considers the domain as a whole,

there are other factors at play that break the symmetry (that is, explain why

partial isomorphisms are not symmetrical) (Bueno & French 2012, pg. 887).

The Argument from Misrepresentation

This argument comes in two kinds, each of which involves two further types of

misrepresentation. The first is mistargeting. This occurs when there is a prob-

lem with the relationship between the target and vehicle. The first type, the

form of mistargeting I will call ‘mistaken target’, occurs when we “mistakenly

suppose the target of a representation to be something that it actually does

not represent” (Suárez 2003, pg. 233). The second type occurs when the target

of the representation does not exist. The second sort of misrepresentation is

the phenomenon of inaccuracy. This can occur either due to representations

leading to empirically inadequate results, or when a representation includes ab-

stractions and idealisations. Misrepresentation due to empirically inadequate

results is not a specific problem for accounts of representation. Rather it is

a general problem for philosophy of science. It is related to issues concerning

measurement, accuracy, and other practical aspects of science. The claim is

that the source of these inaccuracies is not the representation, but the informa-

tion fed into the representation: the data, the experimental results, etc. Any

source of inaccuracy due to the representation itself will fall under the banner

58

of abstraction and idealisation.

The idea behind the abstraction and idealisation type of inaccuracy is that

the representation vehicle does not accurately represent the target due to omit-

ting details (abstraction) or due to claiming something that is not true of the

target (idealisation). Suárez argues that similarity accounts can account for

abstraction and idealisation as such accounts only require the sharing of some

properties between target and vehicle. He also argues that the simple isomor-

phic version of structural accounts cannot account for this kind of misrepre-

sentation, as any difference between target and vehicle would result in there

being no isomorphism between the target and vehicle. Fortunately, there are

structural relation accounts that make use of other structural relations than

isomorphisms and so can potentially avoid this problem. I will address this

type of misrepresentation in more detail in chapters 6 and 7, focusing on how

these accounts can explain the relationship between the faithfulness and use-

fulness of representations, which is a crucial part of understanding how these

accounts fit into Contessa’s approach.

As an example of mistaken target misrepresentation, Suárez asks us to

imagine a friend dressed up as the subject of a painting, such as a painting

of Pope Innocent X. Here, we might mistake the painting to be representing

our friend, rather than the actual Pope Innocent X. Suárez states that the

main reason for the misrepresentation is the ignorance of “the history and the

true target of the representation” (my emphasis) (Suárez 2003, pg. 234). But

this isn’t simply a statement that the right causal history has to be in place

for a vehicle to represent its target, as, for Suárez, the existence of a relation

between vehicle and target does not produce a representation. Rather the

representation has to be used by an agent, and the agent must be capable of

using it: “the skill and activity required to bring about the experience of seeing-

in (the appreciation by an agent of the ‘representational’ quality of a source), is

59

not a consequence of the relation of representation but a condition for it”. The

point Suárez is attempting to make is that due to this essential involvement

of the agent in establishing a representation, the mere existence of a relation

(whether one of similarity or isomorphism) is not sufficient for a representation,

nor is it sufficient for explaining why a representation fails. The explanation for

the failure here is that the agent has only used extant relations of similarity (or

isomorphisms) between the painting and their friend to establish the target of

the painting, when the agent should have also paid attention to other relations

between the painting and potential targets to establish what the painting is a

representation of.

The above example is for artistic representation; Suárez also provides one

for scientific representation. He points to the specific case of a mathematician

providing a solution to a equation, which the mathematician is unaware is the

quantum state diffusion equation. The general point Suárez is attempting to

make is that on the isomorphism account, when “a [mathematician discovers]

a certain new mathematical structure”, if this new mathematical structure is

“isomorphic to a particular phenomenon [this] would amount to the discovery

of a representation of the phenomenon - independently of whether the math-

ematical structure is ever actually applied by anyone to the phenomenon”

(Suárez 2003, pg. 243). This should not be the case: our intuition is that the

mathematical structure only becomes a representation when it is applied to

the quantum phenomenon.

There are two problems with this example as a case of mistaken target. The

first problem concerns the way Suárez has introduced the isomorphism and

similarity accounts as attempts to ‘naturalise’ representation. The second is

due to a disanalogy between the artistic and scientific examples. The point that

a structural relation is insufficient for representation is admitted by proponents

60

of actual structural relation accounts.14 Yet critics of these accounts often

make the mistake of claiming that structural accounts do hold the existence

of structural relations between targets and putative vehicles to be sufficient.

This mistake is based on the belief that structural relation accounts attempt

to ‘naturalise’ representation. Suárez states this belief as follows (Suárez 2003,

pg. 226):

One sense in which we may naturalize a concept is by reducing it to facts,

and thus showing how it does not in any essential way depend upon

agent’s purposes or value judgements (Putnam (2002), van Fraassen

(2002)). The two theories that [Suárez criticises] are naturalistic in

this sense, since whether or not representation obtains depends on facts

about the world and does not in any way answer to the personal pur-

poses, views or interests of enquirers.

Given the number of times that proponents of the structural relations accounts

discuss pragmatic, context and heuristic concerns and the role of agents’ in-

tentions in representation this claim can be seen to not only be false, but also

destructive to the debate. By claiming that the isomorphic type accounts hold

this view, Suárez is arguing against a straw man position.

The disanalogy between the painting example and the quantum state dif-

fusion equation originates in a rejection of the straw man version of the iso-

morphism account. In the state diffusion case, Suárez argues (incorrectly)

that structural relation accounts hold there to be a representation due to the

isomorphism between the solution the mathematician produces and the phe-

nomenon, irrespective of agents using the solution as a representation. The

‘mistargeting’ here is that the solution is not targeted at anything, yet it ap-

pears as though it should be on the (straw man) isomorphism account. In

the painting case, the mistargeting is due to incorrectly identifying the target

14e.g. Bueno & French (2012).

61

of an actual representing vehicle. The painting already has a target, but the

agent has mistakenly identified the target. By rejecting the straw man version

of the isomorphism account, we see that there is no target for the new solu-

tion to be related to. There is no targeting occurring, never mind a case of

mistargeting. A better example would be a case where we had a solution to

the state diffusion equation obtained by a physicist for one particle, but we

mistakenly use the solution to describe a second particle. Here the solution is

being used as a representation, and so there is a target and another putative

target to mistakenly identify. What these two examples, spelt out in this way,

shows, however, is that the cause of the mistargeting in the mistaken target

case is the agent using the representation. They choose the wrong target. The

structural or similarity relations might contribute to this, but as these relations

are not held to be sufficient for representation, the error lies in another part of

the account (i.e. the intention of the agent who is using the representation).

This argument shows that the cause is not the structural relation, and so the

problem of mistaken target is not a special problem for structural accounts

of representation. The argument also shows that the fault in such cases lies

with the agents and their use of the representation vehicles. This solves the

problem.

The final type of mistargeting occurs when the target does not actually

exist. This has occurred frequently in the history of science. Examples include

phlogiston and the mechanical ether. Initially this seems like a large problem.

It asks the question of how we can hold there to be a representation of a target,

when that target does not exist? I will answer this question below, in §3.5, in

response to Chakravartty’s discussion of the issue.

62

The Non-necessity & Non-sufficiency Arguments

As we have seen above, the straw man version of the isomorphism account

claims that the existence of an isomorphism between vehicle and target estab-

lishes a representation. Thus any extant isomorphism is sufficient for repre-

sentation. What is missing in this case, according to Suárez, is an intention

for the vehicle to represent the target. Non-sufficiency is clearly accepted by

structural relation proponents: for example Bueno and French admit that this

is the case, and claim that other factors combine with the structural relation

to be jointly sufficient, where these factors are “broadly pragmatic having to

do with the use to which we put the relevant models [representations]” (Bueno

& French 2012, pg. 887).15

With respect to the non-necessity argument, anyone who admits a non-

uniform account of representation (in general) will be able to accept the non-

necessity of any type of relation to establishing representation. Perhaps Suárez’

point could be restricted to just scientific representations. This might be what

Suárez intends given that he advocates abandoning the search for “universal

necessary and sufficient conditions that are met in each and every concrete real

instance of scientific representations” (Suárez 2004, pg. 771). In response to

this charge, advocates of structural relation accounts and similarity accounts

admit the necessity of their preferred relation. The argument is straightfor-

ward: without such relations, the success of scientific representations would be

a miracle.16

15A similar move is available to the similarity account. As I explained above, Giere arguesfor representation being impossible without agents, which implies that he takes similarityrelations to be insufficient for representation (Giere 2006, pg. 60, 64 endnote 13).

16Bueno & French (2012) endorse Chakravartty’s (2009) argument along these lines(though they endorse structural relations while Chakravartty endorses a similarity relation).They go on to discuss the need for such a relation to ground the inferences that scientificrepresentations facilitate, a position I adopt and explore after discussing Contessa’s position.

63

Further Arguments, and Amended & Weakened Versions

To be fair to Suárez, he does recognise that there are versions of the similarity

and isomorphism/structural relation accounts that do not attempt to natu-

ralise representation. He describes these as amended versions, accounts that

employ a notion of ‘representational force’, which he defines as (Suárez 2003,

pg. 237):

A’s capacity to lead a competent and informed enquirer to consider

B as the representational force of A . . . [Representational forces] are

determined at least in part by correct intended uses, which in turn are

typically conditioned and maintained by socially enforced conventions

and practices: A can have no representational force unless it stands in a

representing relation to B; and it cannot stand in such a relation unless

it is intended as a representation of B by some suitably competent and

informed inquirer.

These amended versions avoid the non-sufficiency argument. The logical argu-

ment might also be avoided. Suárez’ formulation of the amended versions of

the structural relation and similarities accounts involves the additional condi-

tion of A’s representational force pointing at B. The avoidance of the logical

argument depends upon how one explicates intended use. The other argu-

ments, however, are claimed to continue to apply. While the accounts I will

look at do not adopt Suárez’ notion of representational force, their authors

do discuss the intentions of representing agents, and the relationship between

representations and intentions on their accounts.

Suárez also recognises that there are structural relation accounts that em-

ploy other structural relations than isomorphisms. He discusses the use of ho-

momorphisms and partial isomorphisms as possible representation relations,

though he finds them wanting. Homomorphisms are claimed to be able to cope

with “partially accurate models”, and so avoid the abstraction and idealisation

64

version of the argument from misrepresentation. It is also held to weaken

the non-necessity argument (though I do not take this argument to have any

force), but suffers from the other arguments. In particular, homomorphism

does not avoid the logical argument due to being reflexive. Fortunately, one

might leverage the notion of representational force (or similar) to avoid this

problem.

Possible Solutions to the CQ

I take the above discussion to show that the main arguments against the struc-

tural relations account are not successful against suitably refined versions of

such accounts, such as those I will outline in the next chapter. I do take the

arguments to tell against an account of similarity, however. Thus in order to

answer the constitution question we are left with a choice between Suárez’ ac-

count of representation and a structural relation account. Suárez’ account of

representation is explicit in not attempting to establish necessary conditions,

in particular in claiming that a particular type of relation is necessary for rep-

resentation (Suárez 2004, pg. 771). He formulates his Inferential Conception

Approach (ICA) as:

[Inf] A represents B only if:

(i) the representational force of A points towards B, and

(ii) A allows competent and informed agents to draw specific inferences

regarding B.

As such, it does not make any sense to ask whether the ICA provides an answer

to the CQ; it is compatible with any putative representation relation, provided

that the relation can, for the representation in question, allow for the “specific

inferences” to be drawn.

Thus in order to argue for the structural relation accounts, one would have

to either: a) show that structural relations are the best candidates for the

65

representation relation in all instances of scientific representation (i.e. it is the

relation that provides the inferences in the most accessible or efficient way); or

b) that structural relations are the only relation that provide the inferences.

a) is obviously weaker than b), and b) requires a response to the non-necessity

argument (which may be restricted to scientific representation). Suárez’ po-

sition and my counter argument rest on the assumption that inferences play

an essential role in scientific representation. Such an assumption might form

part of, if not the, answer to the DQ. Thus I will now turn to the DQ and

argue that the establishment of surrogative reasoning is a distinctive feature

of scientific representation.17

3.3.2 The Demarcation Question

The DQ, as set out above by CC, asks “what is distinct about scientific repre-

sentation?” CC’s answer is as follows (Callender & Cohen 2006, pg. 83):

Plausibly scientific representation is just representation that takes place

when the agents are scientists and the audience are either fellow scientists

or the world at large.

What this means for CC is that the demarcation problem is transformed from

concerning scientific representation to science itself: it is “ the demarcation”

problem of science. CC reach this conclusion because they believe that all

representation can be reduced to a form of mental representation. Thus there is

nothing special about scientific representation other than the fact that it occurs

exclusively within the domain of science, or that it is a form of representation

performed by scientists.

The issue then, is whether there is anything unique about scientific repre-

sentation that prevents it from being reduced to some fundamental represen-17At least this is a distinctive feature of epistemic representation, of which scientific

representation is a subspecies.

66

tation (as CC presume there is not). Above I claimed that this unique feature

would turn out to be the ability to draw inferences. Further, to prevent reduc-

tion to another type of representation, it appears that these unique features

must be provided by the representation relation required by the particular type

of representation. i.e. that scientific representation has the unique feature of

allowing inferences to be drawn about the target by the representing vehicle,

and these inferences are dependent upon a structural relation from the target

to the vehicle.

A challenge to taking such surrogative inferences to be distinctive of scien-

tific representation can be found in Contessa (2011), where Contessa argues for

a demarcation between epistemic and non-epistemic representations based on

surrogative inferences. While this distinction is complimentary to my project,

it actually cuts across the demarcation that CC discuss and forces me to agree

with their conclusion but for different reasons. I will outline Contessa’s ar-

guments for epistemic representations in the next section and argue that one

should answer the DQ in terms of epistemic representation performed by sci-

entists.

3.4 Contessa’s Approach

Contessa is initially concerned with the demarcation between epistemic and

non-epistemic representations, and later between merely epistemic and faithful

epistemic representation. Epistemic representation is characterised by Con-

tessa as follows (Contessa 2007, pg. 52-53):

A vehicle is an epistemic representation of a certain target for a certain

user if and only if the user is able to perform valid (though not necessarily

sound) surrogative inferences from the vehicle to the target.

and as follows (Contessa 2011, pg. 123, endnote 7):

67

To say that a representation is an epistemic representation is just to say

that it is a representation that is used for epistemic purposes (i.e. a

representation that is used to learn something about its target)

An example of non-epistemic representation is aesthetic representation, i.e.

how a painting represents its subject. These categories of representation are

not mutually exclusive, in that an epistemic representation can also be un-

derstood to be an aesthetic representation. Contessa’s concern is the main

purpose of the representing vehicle here, so talks in broad terms about epis-

temic or aesthetic representation. Thus establishing something as an epistemic

representation does not prevent it from being an aesthetic representation - one

can clearly gain information from a painting - yet the main purpose of the

painting is aesthetic, so in broad terms the painting is taken to be an aesthetic

representation.

Contessa takes the main symptom of epistemic representation to be the

ability of agents to perform surrogative reasoning about the target of the rep-

resentation by the representational vehicle. A key feature of epistemic repre-

sentations is that the surrogative reasoning does not need to provide sound

inferences about targets in order for the vehicles to be successful epistemic

representations. The difference between the soundness and unsoundness of

the inferences can be cashed out in terms of the faithfulness of the represen-

tations (Contessa 2007, pg. 54). A completely faithful representation is one

where all the surrogative inferences we can draw from the vehicle to the target

are sound. A partially faithful representation is one that provides some sound

inferences from the vehicle to the target, but not all inferences. A partially

faithful representation can still be a successful representation.18 Contessa uses

this distinction between merely epistemic and faithful epistemic representation

to pose two questions that he thinks an account of representation should be

18These are the concepts I will adopt and use to answer the NI.

68

able to answer:

I) What makes a vehicle an epistemic representation of a certain target?

II) What makes a vehicle a more or less faithful epistemic representation of

a certain target?

He also claims that these questions have been conflated into a single question

concerning the ‘problem of scientific representation’.

It is here that Contessa’s distinction between epistemic and non-epistemic

representation comes into direct conflict with the DQ. This is made most ex-

plicitwhen he states: “another way in which the label ‘scientific representation’

can be misleading is that it seems to imply that something sets aside sci-

entific representations from epistemic representations that are not scientific”

(Contessa 2011, pg. 124, footnote 10). The task at hand, then, is to look

at Contessa’s arguments for his preferred account of epistemic representation

and faithful epistemic representation, and see whether he establishes epistemic

representation as distinct from, or encompassing, scientific representation.

Contessa claims that there are three rival accounts epistemic representa-

tion: the denotation account; Suárez’ ICA; and the Interpretational Account.

He also holds there to be two (“some related”) accounts of faithful epistemic

representation: the similarity account and the structural account.

Contessa’s project can understood as the attempt to answer the following

questions:

1. What are, if any, the necessary and sufficient conditions for epistemic

representation?

2. Do the necessary and sufficient conditions for epistemic representation

have any necessary and sufficient conditions themselves?

3. What are, if any, the necessary and sufficient conditions for faithful epis-

temic representation?

69

Contessa answers 1 and 2 in his (2007) and claims that structural relations will

play a role in answering 3 in his (2011). He is clear that the first two questions

can be separated from the third, in that the questions of how to establish

epistemic representation and how to establish faithful epistemic representation

are two distinct questions and should be addressed separately (Contessa 2007,

pg. 67-68), (Contessa 2011, pg. 124). This is partly due to their different

nature: epistemic representation is binary, either we have it or we do not,

whereas faithful epistemic representation is gradable (Contessa 2007, pg. 55).

3.4.1 Necessary & Sufficient Conditions for Epistemic

Representation

Contessa argues for denotation to be a necessary condition for epistemic repre-

sentation but not a sufficient condition (Contessa 2011, pg. 125). An example

Contessa uses is the use of an elephant to represent the London Underground.

While one can denote the London Underground via the elephant, it is not clear

how one could use the elephant to perform surrogative inferences about the

network. This example indicates that there are further conditions to satisfy in

order to have a case of epistemic representation.

The ICA can be understood to be providing the second necessary condition

for an epistemic representation. For example in Suárez (2004) the necessary

condition is given as an agent being able to perform surrogative influences from

the vehicle to the target. Contessa believes that this approach is “ultimately

unsatisfactory” and in fact involves an ad hoc move: the ICA “seems to turn

the relation between epistemic representation and surrogative reasoning upside

down” (Contessa 2011, pg. 125). Take the example of the map of the London

Underground. The ICA suggests that the map represents the Underground

network in virtue of the fact that you can perform surrogative inferences from

it to the network. However, Contessa argues, the reverse seems to be the case:

70

one can perform surrogative inferences to the network from the map in virtue

of the fact that the map is an epistemic representation of the network. i.e.

the ability to draw surrogative inferences depends on a map being epistemic

representation, not the other way around. “If you do not take [the map] to be

an epistemic representation of the London Underground network in the first

place, you would never try to use it to perform surrogative inferences about

the network.”

Contessa’s position on the ICA can be summarised as the view that an ac-

count of epistemic representation should explain what makes a certain vehicle

into an epistemic representation of a certain target (for a certain user) and

how in doing so it enables the agent to perform surrogative inferences about

the target. The ICA, however, takes the ability to draw surrogative inferences

to be basic and so lacks this explanation.

Given the initial characterisation of an epistemic representation19 and the

above, it is clear that a necessary and sufficient condition for epistemic repre-

sentation is that a vehicle allows an agent to perform valid surrogative infer-

ences about the target. This allows question 2 to be rephrased as follows:

2′ What are the necessary and sufficient conditions for an agent to be able

to use a vehicle to draw valid surrogative inferences?

I will outline Contessa’s answer to 2′ in the next section.

3.4.2 Necessary & Sufficient Conditions for Surrogative

Inferences

In order to provide an account of epistemic representation, Contessa proposes

his Interpretational Account, where a vehicle is an epistemic representation

19“A vehicle is an epistemic representation of a certain target for a certain user if andonly if the user is able to perform valid (though not necessarily sound) surrogative inferencesfrom the vehicle to the target” (Contessa 2007, pg. 52-53).

71

of a certain target for a certain user (i.e. a user can use the vehicle to draw

valid surrogative inferences) if and only if: the user takes the vehicle to denote

the target; and the user adopts an interpretation of the vehicle in terms of the

target.20 The DDI account of Hughes (1997)21 is selected as a possible reference

for an account along similar lines. The key feature of the Interpretational

Account for the present discussion is that “the adoption of an interpretation of

the vehicle in terms of the target [is] what turns a case of mere denotation into

one of epistemic representation” (Contessa 2011, pg. 126). The interpretation

of the vehicle in terms of the target supposedly provides the user with a set

of “systematic rules to ‘translate’ facts about the vehicle into (putative) facts

about the target”. This set of systematic rules is the Interpretational Account’s

explanation of the relation between epistemic representation and surrogative

reasoning. I will now outline the Interpretation Account and draw out the

necessary and sufficient conditions it fulfils for surrogative inferences.

Contessa argues for the following (Contessa 2007, pg. 57):

A vehicle is an epistemic representation of a certain target (for a certain

user) if and only if the user adopts an interpretation of the vehicle in

terms of the target.

He takes the adoption of an interpretation to be what grounds the ability to

perform surrogative inferences from vehicle to target and so is the necessary

and sufficient condition for obtaining an epistemic representation. His argu-20The story is slightly more complicated however. To avoid accusations of ‘naturalising’

representation, Contessa points out that a vehicle is an epistemic representation for a user,rather than an epistemic representation “in and of itself” (Contessa 2007, pg. 53). Contessathus claims that the representation relation in the case of epistemic representation is atriadic relation between the vehicle, a target and a (group of) user(s). Above, in §3.3.1 -The Argument from Misrepresentation, I mentioned this way of avoiding a naturalisationof representation and responding to the non-sufficiency argument. I have yet to providedetails for handling intentions - I will do this when I discuss the accounts in detail in thenext chapter. Contessa opts for one of the possible ways to accommodate the intentions ofagents: he includes the agent as a relata of the (epistemic) representation relation. Not allof the accounts follow this approach.

21This account is a major influence on the structural relation account of Bueno & Colyvan(2011) and Bueno & French (2012).

72

ments for this position consist in cashing out the notion of interpretation, and

showing how adopting an interpretation of the vehicle in terms of the target

is necessary and sufficient for the performance of surrogative inferences from

vehicle to target.

A loose first pass of cashing out the notion of interpretation, what I will

call the ‘general notion’, is that a user “interprets a vehicle in terms of a target

if she takes facts about the vehicle to stand for (putative) facts about the

target”. This can be considered a good starting point, which we can tighten

up in certain respects to obtain quite specific and detailed characterisations of

interpretation. One such characterisation is an analytic interpretation. This

is a set-theoretic understanding of interpretation. It proceeds by the user first

identifying the following (Contessa 2007, pg. 57-58):

• A (nonempty) set of relevant objects in the vehicle: ΩV = oV1 , . . . , oVn

• A (nonempty) set of relevant objects in the target: ΩT = oT1 , . . . , oTn

• A (possibly empty) set of relevant properties of and relations among

objects in the vehicle: P V = nRV1 , . . . ,

nRVm, where nR denotes an n-ary

relation; properties are construed as 1-ary relations.

• A set of relevant properties and relations among objects in the target:

P T = nRT1 , . . . ,

nRTm

• A set of relevant functions from (ΩV )n to ΩV (ΨV = nF V1 , . . . ,

nF Vm),

where nF denotes an n-ary function, and (ΩV )n is the Cartesian product

of ΩV with itself n times.

• A set of relevant functions from (ΩT )n to ΩT (ΨT = nF T1 , . . . ,

nF Tm)

A user must then satisfy the following conditions:

1. The user takes the vehicle to denote the target;

73

2. The user takes every object in ΩV to denote one and only one object in

ΩT and every object in ΩT to be denoted by one and only one object in

ΩV ;

3. The user takes every n-ary relation in P V to denote one and only one

relevant n-ary in P T and every n-ary relation in P T to be denoted by

one and only one n-ary relation in P V ;

4. They take every n-ary function in ΨV to denote one and only one n-ary

function in ΨT and every n-ary function in ΨT to be denoted by one and

only one n-ary function in ΨV .

His in (2011), Contessa opts for a slightly different description of the sit-

uation. He states that the interpretational account has two necessary and

sufficient conditions: the user takes the vehicle to denote the target; and the

user adopts an interpretation of the vehicle in terms of the target (Contessa

2011, pg. 126). This might seem a different set of conditions, as he takes a

denotation relation to a separate condition to interpretation. The difference,

however, is only one of bookkeeping; the same relations will still be present,

just that in his (2007) version of the Interpretation Account the denotation

relation will be included under a composite interpretation relation, whereas in

the (2011) version it is separated from the composite interpretation relation.

Both relations are part of the composite representation relation in either case.

As evidence of this being an insignificant difference, I point to Contessa’s ex-

planation of how one might understand the Rutherford model of the atom to

be an epistemic representation. He states that on the interpretation account,

the Rutherford model is an epistemic representation of an atom if and only if

(Contessa 2007, pg. 59):

(1) they [the user] take the model as a whole to stand for the atom in

question . . . , (2) they take some of the components of the model to stand

74

for some of the components of the system, and (3) they take some of the

properties of and relations among the objects in the model to stand for

the properties of and relations among the corresponding objects in the

system (if those objects stand for anything in the atom).

It appears that Contessa separates out the relation of denotation between vehi-

cle and target (call this the ‘global denotation relation’) in order to emphasise

that the vehicle has to be aimed the a particular target we use to use the

vehicle for in order to obtain valid inferences. He explains that, without the

global denotation relation, one might still be able to draw inferences from a

vehicle to a target due to some general features of the vehicle. For example,

subway maps are designed with a general interpretation in mind, and so can be

epistemic representations for subway networks other than the ones they were

explicitly designed for (Contessa 2011, pg. 126, endnote 14).

So far, so good, but in order to properly explain how adopting such an

interpretation allows one to draw valid surrogative inferences from a vehicle

to a target something implicit in this characterisation of interpretation needs

to be made explicit. That is, adopting an analytic interpretation results in the

adoption of a set of inference rules (Contessa 2007, pg. 61):

Rule 1: If oVi denotes oTi according to the interpretation adopted by

the user, it is valid for the user to infer that oTi is in the target if and

only if oVi is in the vehicle;

Rule 2: If oV1 denotes oT1 , . . . , oVn denotes oTn , and nRVk denotes nRTk

according to the interpretation adopted by the user, it is valid for the

user to infer that the relation nRTk holds among oT1 , . . . , oTn if and only if

nRVk holds among oV1 , . . . , oVn

Rule 3: If, according to the interpretation adopted by the user, oVi de-

notes oTi , oV1 denotes oT1 , . . . , o

Vi denotes oTi , and

nF Vk denotes nF Tk , it

is valid for the user to infer that the value of the function nF Tk for the

75

arguments oT1 , . . . , oTn is oTi if and only if the value of the function nF Vk

if oVi for the arguments oV1 , . . . , oVn .

An inference is then valid if it is in accordance with Rule 1, 2 or 3.

So much for cashing out of the notion of interpretation and providing an

explanation for why it provides valid surrogative inferences from vehicle to tar-

get. Contessa still has to provide arguments for why adopting an interpretation

is necessary and sufficient. As Contessa explicitly states that he does not want

to give the impression that all interpretations are necessarily analytic, he only

provides arguments for why adopting an analytic interpretation is sufficient

for epistemic representation (Contessa 2007, pg. 58, 63). I therefore take his

argument for adopting an interpretation to be necessary for surrogative infer-

ences to be the claim that one must take facts about the vehicle to stand for

(putative) facts about the target in order to infer anything from the vehicle to

the target (i.e. the general notion).

Contessa argues for the sufficiency of the analytic interpretation by look-

ing for a case where a user adopts an analytic interpretation of the vehicle yet

cannot draw valid inferences about the target, i.e. the vehicle fails to represent

(qua epistemically represent) the target. Key to understanding his arguments

that such situations do not exist is the recognition of the distinction between

mere epistemic representation and faithful epistemic representation. Most im-

portantly, that epistemic representation requires only valid inferences, while

faithful epistemic representation requires sound inferences, where soundness is

cashed out in terms of the inferences being valid and their conclusions true of

the target (Contessa 2007, pg. 51).22

22Contessa defends his account by showing how the Rutherford model of the atom canbe understood as an epistemic representation of a hockey puck sliding on the frozen surfaceof a pond (Contessa 2007, pg. 64-65). These arguments are informative, but do not need tobe summarised here.

76

3.4.3 Faithful Epistemic Representation

The account of faithful epistemic representation that Contessa favours is a

structural (relation) account. Contessa presents a characterisation of such an

account as follows: if a vehicle is an epistemic representation of a target for

a certain user, then, if some specific morphism holds between the structure of

the vehicle and the structure of the target, then the model is a faithful rep-

resentation of that target. This seems to imply that Contessa takes epistemic

representation to involve a denotation relation and an interpretation relation,

and faithful epistemic representation to involve a relation of denotation, of

interpretation, and a structural relation.

After this review of Contessa’s arguments concerning epistemic and faith-

ful epistemic representation, one may ask the following questions. Can a clear

distinction be drawn between scientific and epistemic representation? If not,

is scientific representation a subspecies of epistemic representation? And if it

is, is scientific representation identical with faithful epistemic representation?

If not, what distinguishes scientific representation from faithful epistemic rep-

resentation?

3.4.4 Scientific and Epistemic Representation

In the discussion above concerning the CQ I noted that Suárez’ position and

my (future) arguments in favour of the structural relations accounts of scien-

tific representation rested (and would rest) on the assumption that establishing

surrogative reasoning was a unique feature of scientific representation. As I

also noted above, this assumption is a solution to the DQ. However, the pre-

ceding review of Contessa shows this assumption to be unjustified; surrogative

reasoning appears to be a unique feature of the broader category epistemic

representation. Contessa frequently makes use of the example of a map of

77

the London Underground. There can be no denying that a map is a repre-

sentational vehicle that allows one to gain information about a target, and to

gain information through surrogative reasoning. Yet any attempt at calling a

map a scientific representation would undermine the term ‘scientific represen-

tation’ if it is to have any special meaning at all. I therefore take Contessa

to have shown that surrogative reasoning is not a unique feature of scientific

representation, but of epistemic representation.

Surrogative reasoning might be a necessary but insufficient condition for

scientific representation. I therefore take scientific representation to be a sub-

species of epistemic representation. Thus there is no clear distinction to be

drawn between epistemic and scientific representation.

The example of the Underground map can be used to answer the third

and fourth questions. An old map of the Underground that does not have

modern stations or lines on it is a less faithful representation of the modern

Underground than a modern map. However, one can still draw some true

conclusions about the Underground from the older map. Similarly, scientific

representations of varying faithfulness can be used to draw true conclusions

about their targets (Contessa discusses this with the example of the model

of the toboggan with differing gravitational sources included in the model).

From just this brief description of different examples of varyingly faithful epis-

temic representations it should be clear that simply because a representation

is partially faithful has nothing to do with whether it is classed as scientific or

not. Thus the answer to the third question (is scientific representation iden-

tical with faithful epistemic representation) is negative because we can have

partially faithful, non-scientific representations.

A definite answer to the fourth question (what distinguishes scientific rep-

resentation from faithful epistemic representation) is challenging to give. It

is not clear that all cases of scientific representation are cases of faithful epis-

78

temic representation. It might be the case that all scientific representations are

partially faithful, in that they facilitate at least one sound inference. It is not

immediately clear how to settle this issue. For any representation to be useful

in any way it must facilitate at least one sound inference about its target. The

issue is how to evaluate whether scientific representations that are typically

taken to be completely inaccurate representations give any sound inferences.

Examples of such representations are those that suffer from non-existent tar-

gets, such as the mechanical ether. It is reasonable to take such representations

as providing some sound inferences, namely empirically adequate predictions

at a certain time, even though any inferences concerning the ontology of the

target are completely unsound. I will discuss this issue in more detail in the

next section.

It is therefore the case that we cannot answer the DQ by only pointing to

surrogative reasoning. This is a distinguishing feature of epistemic represen-

tation, and while all scientific representation is at least epistemic representa-

tion, not all epistemic representations are scientific ones. I suggest that the

DQ should be answered in a similar way to CC: what distinguishes scientific

representation from other forms of representation is that it is epistemic rep-

resentation performed by agents who are scientists, whose audience is other

scientists or the world. The above discussion has gone a good way to showing

that this is a suitable answer.

Before turning to answer the CQ with this answer to the DQ in mind, I

will survey Chakravartty’s discussion of what he calls informational and func-

tional accounts of representation.23 His discussion at first appears supportive

of Contessa’s approach and the answer to the DQ I wish to endorse. However,

he raises an issue that he claims requires further investigation than he is able

to provide in his paper. I attempt this further investigation below, and argue

23Chakravartty (2009).

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that Contessa’s approach fits in with his earlier arguments, and that the issues

can be settled via Contessa’s approach.

3.5 Chakravartty: Informational vs. Functional

Accounts of Representation

Chakravartty reviews a different way of answering the DQ, where the DQ is

phrased as “what . . . are the ‘essential’ properties of a scientific representa-

tion?” (Chakravartty 2009, pg. 198). He considers there to be two, supposedly

conflicting, categories into which one can place the various attempts to answer

this question that can be found in the literature. He names these categories

informational and functional. Accounts that can be classed as informational

are those that claim a “scientific representation is something that bears an

objective relation to the thing it represents, on the basis of which it contains

information regarding that aspect of the world”. The most general version

of the informational approach will appeal to relations of similarity. Accounts

to be classed as functional emphasise the functions of representations: “their

[the representations’] uses in cognitive activities performed by human agents in

connection with their targets” (Chakravartty 2009, pg. 199). These functions

can be divided into further categories: the demonstrations and interpretations

of target systems the representations allow; and the inferences they permit

concerning aspects of the world.

In his discussion of this distinction, Chakravartty highlights three argu-

ments, “charges”, against the informational accounts that are intended to show

the superiority of the functional accounts. These are a (familiar) charge of

non-necessity, a (familiar) charge of non-sufficiency, and the charge that there

are essential functions to scientific representation that informational accounts

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remain silent about and are therefore “defective”.24 Chakravartty deals with

the third charge by arguing that this worry is predicated on the confusion

between the means and ends of scientific representation. To summarise his

argument: informational and functional theories focus on different aspects of

the question ‘what are scientific representations?’ Informational theories con-

sider representations as knowledge bearing entities, whereas functional theories

consider representations to be a set of knowledge exercising practices. These

different conceptions of representation are not contradictory; in fact they are

complementary, thus questions asked of these two conceptions require answers

that must be taken together to form a general understanding of scientific rep-

resentation.

I agree with Chakravartty’s responses and accept his arguments to all of

the charges. I also agree that the dichotomy between informational and func-

tional accounts of scientific representation is a false one. The natural next

step, given the rejection of the dichotomy, would be to endorse an account of

scientific representation that had features of both the informational and func-

tional accounts, namely intentionality, a similarity or structural relation, and

surrogative reasoning. One such approach is Contessa’s. While Chakravartty

is willing to entertain this idea, he cautions against it being straightforward.

There is an issue that requires further investigation before this kind of account

can be adopted without difficulty.

The issue concerns cases of mistargeting, where the target does not ex-

ist such as in the case of phlogiston or the mechanical ether, and accounts

of representation that make use of a gradable notion of accuracy or success.

24I have briefly covered Chakravartty’s response to the non-necessity charge above in§3.3.1 - The Non-necessity & Non-sufficiency Arguments. He appeal’s to Goodman’s claimthat similarity is “no sufficient condition representation” (Goodman 1976, pg. 3-4) and pointsout that it does not appear to be part of any informational account that similarity orisomorphism (or any other such relation) is sufficient for representation (Chakravartty 2009,pg. 205). He also appeals to the role of agents’ intentions in supplying the non-symmetryand non-reflexive properties to the representation relation.

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Chakravartty argues that it is unclear how to treat such cases. Take the model

of the mechanical ether. Chakravartty claims that further work has shown it

to be about as inaccurate as it is possible to be. He offers the reader two re-

sponses. One might conclude that the model was an unsuccessful/inaccurate

representation of its intended system (due to the system not existing), yet is

still a scientific representation. Alternatively one might conclude that such a

model is undoubtably scientific (due to its presence in previous scientific inves-

tigations), but due to how unsuccessful/inaccurate it was, it is not a represen-

tation, merely a model (we were mistaken in taking it to be a representation)

(Chakravartty 2009, pg. 209). Chakravartty takes this problem of mistargeting

due to the non-existence of the target to be significant, as whether we count

models which mistarget in this way will determine the stance one takes with

respect to the necessary and sufficient conditions for scientific representation.

For example, if one accepts a distinction between mere representation and ac-

curate representation, then one can exclude the information relation from the

necessary conditions for mere representation and include in the necessary con-

ditions for accurate representations. However, if one rejects this distinction,

then one might hold that there is some threshold of accuracy for a represen-

tation to count as scientific, and hence maintain that the information relation

is a necessary condition for scientific representation.

I take Contessa’s distinction between epistemic and faithful epistemic rep-

resentation to fit the distinction between mere and accurate representation. As

outlined above, Contessa’s approach does entail different necessary and suffi-

cient conditions for epistemic (mere) representation from faithful epistemic

(accurate) representation. We can now decide on whether scientific represen-

tation should be a subspecies of epistemic or faithful epistemic representation:

I have already argued that we should consider scientific representation to be a

subspecies of epistemic representation. The issue is whether all scientific rep-

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resentations can be considered partially faithful representations. Remember

that even one sound surrogative inference allows us to call a representation a

partially faithful representation.

Accepting that all scientific representations are epistemic representations

allows us to call cases like phlogiston and the mechanical ether scientific

representations: they are epistemic representations performed by scientists.

Chakravartty anticipates such a response to his worry, in that he describes

‘scientific representation’ as a “term of art”, such that “we may define it as

best serves the various philosophical uses to which it is put”, defending this

conventional approach against a definitive approach by pointing to conflicting

intuitions (Chakravartty 2009, pg. 210). I personally hold the intuitions that

misrepresentations are still representations, and that the term ‘scientific rep-

resentation’ does not connote anything distinctive. I believe that Contessa’s

approach provides a justification for these positions, contra the ideas that there

is there is a minimum level of accuracy for a representation to count as sci-

entific and that a putative representation’s target has to exist for it to be an

representation.25

Part of how Contessa’s approach provides this justification is the separa-

tion of faithfulness and success (or usefulness) (Contessa 2011, pg. 130). By

grounding faithfulness in terms of the soundness of the surrogative inferences,

we can understand faithfulness as involving the information relation. Indeed,

Contessa points to this himself, arguing for a structural relation account of

faithful epistemic representation. By grounding success in heuristic, contex-

tual and pragmatic terms, we can have successful representations that are

wildly unfaithful and unsuccessful representations which are very faithful. A

25In the next chapter, I will be arguing that Pincock’s Mapping Account from his (2012)fits into Contessa’s approach. Here I wish to note what Pincock says concerning phlogistonas an early indication that his account will fit: he claims that such representations havecontent, such that had phlogiston existed their representations would have been correct.Due to phlogiston not existing, however, their representations all turned out to be incorrect(Pincock 2012, pg. 26).

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further advantage can be found by looking at two different ways in which we

might evaluate the success of the mechanical ether representation. We might

evaluate the representation along instrumentalist lines or considering whether

it manages to ‘save the phenomena’,26 or along ontological lines. The mechan-

ical ether representation had some success in providing accurate predictions,

and so might be considered successful according to the instrumentalist/saving

the phenomena line. However, it failed completely in terms of the ontology

it posited. Contessa’s approach lets us explain why this is the case. The

mechanical ether, as a partially faithful epistemic representation allowed for

some sound surrogative inferences to be drawn concerning the behaviour of

light. However, all surrogative inferences that we could draw from it concern-

ing the ontology of light were found to be unsound. Evaluation of the success

of a representation, then, involves restricting ourselves to certain sets of the

surrogative inferences we can draw from the vehicle, and establishing the num-

ber of sound inferences in that set. The total number of sound and unsound

inferences is not a guide to the success of the representation; we need a context

within which to weigh the number of sound and unsound inferences, and in

this sense success is contextual. The context often comes from the aims of the

user of the representation.

In not recognising that accuracy and success as separate, Chakravartty is

unable to make use of this response to the mistargeting worry. I think that part

of the reason he does not recognise this separation is that he holds an intuition

that scientific representation should be ontologically committing. That is, any

scientific representation should always be evaluated in terms of the ontology it

prescribes. This is made clear in his definition of the informational accounts in

terms of there being an “objective relation” between the vehicle and the target.

26I have in mind something like the distinction van Fraassen draws between the notionsof appearance and phenomena when he briefly discusses the idea of ‘saving the phenomena’(van Fraassen 2010, pg. 8).

84

I want to reject this assumption. I think that we can evaluate a scientific

representation in terms of its ability to save the phenomena separately from

its ontology. Developing this position further, however, will require further

work on what the target of the representation is. A simple reading of scientific

representations would be that the target is the world (indeed, this seems to be

the root of Chakravartty’s complaints that the mechanical ether representation

is inaccurate). A more sophisticated reading will take into account phenomena

and data models, particularly if one adopts the Semantic View of theories.

There is clearly more work to be done here. I will leave this discussion at this

point as it is straying into issues of realism, which are unrelated to my main

argument.

Above I have surveyed Chakravartty’s discussion of the DQ and informa-

tional & functional accounts of representation. I have argued that Contessa’s

approach solves the issue of mistargeting qua non-existent targets Chakravartty

raises for accounts with gradable notions of accuracy. In doing so, I also ar-

gued that Contessa’s approach justifies a rejection of the intuition that there is

anything significant to the term ‘scientific representation’, in that it allows for

one to hold onto the intuition that a misrepresentation is still a representation.

This leads me make my answer to the DQ more precise: scientific represen-

tation is epistemic representation performed by scientists, when their target

is other scientists or the world. However, any successful scientific representa-

tion will require some sound inferences to be drawn. Given the arguments I

put forward concerning misrepresentation and the context dependence of the

notion of ‘success’, it seems plausible that all scientific representations can be

considered partially faithful. Therefore, while the answer to the DQ requires

only that we can draw valid surrogative inferences, our answer to the CQ will

require that we can draw sound surrogative inferences.

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3.6 Conclusion: Answering Callender and

Cohen’s Questions

3.6.1 Answering the Constitutive Question

In this chapter I have argued that structural accounts of representation survive

the counter arguments of Suárez, that scientific representation should be con-

sidered epistemic representation performed by scientists and it is highly likely

that all scientific representations should facilitate at least one sound inference.

I have also argued that the answer to the DQ will set out the conditions that

the representation relation must fulfil. In the case of scientific representation,

the relation must be capable of providing sound surrogative inferences. In

order to do this, the relation must be capable of providing an interpretation

of the representational vehicle in terms of the target, and there must be some

kind of denotation involved. There is no requirement that all of this is achieved

by a single relation: the representation relation can be a compound relation.

For example, Contessa argues for the relation responsible for epistemic repre-

sentation to consist in a denotation relation and the relation(s) responsible for

his analytic interpretation. He then also claims that structural mappings will

constitute the faithful epistemic representation relation.27

I think that Contessa has the right of it: structural relations are the best

candidates for constituting the relation responsible for faithful epistemic repre-

sentation. I also agree that there needs to be some kind of denotation involved,

and that the agents’ intentions need to be accommodated. There is of course

the advantage that adopting a structural relational account of representation

fits with the answer to the applied metaphysical question I provided in the

previous chapter. There I argued that a structural relation could account for

27He also includes the intention of the agents.

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how mathematics can be applied to the world. But I also argued that the

relation would need to be able to account for mathematical representation as

these issues are the more interesting and difficult issues that need answering.

I have shown in this chapter that structural relations are up to this job as

well. How we are able to accommodate the gradable nature of faithfulness

and usefulness will be investigated in the next chapter. Issues that are still

outstanding relating to the problem of misrepresentation due to abstraction

and idealisation will also be addressed in subsequent chapters.

3.6.2 Answering the Normative Issue

The final question to address is the NI, which asks “what is it for a representa-

tion to be correct”? CC introduced this problem while discussing the idea that

models are not truth-apt. This causes a problem for answering this question.

This problem is avoided due to the notion of adopting an interpretation of the

vehicle in terms of the target, and by evaluating the soundness of the surrog-

ative inferences we draw via the representation. This means that we don’t

discuss the representation being ‘true’, but rather of it being faithful, partially

faithful or entirely unfaithful. We can judge the conclusions of our inferences

as being true or false of the target, as they will be fully interpreted statements

about the target. For example, a numerical prediction from a mathemati-

cal representation will actually be a statement about the physical value of a

measurement, not a statement about the representation. Again, however, the

specifics of how to answer this question will depend on the account to hand.

A full answer to this question will therefore have to wait until later chapters.28

I will now turn to setting out two accounts of scientific representation that

employ structural relations as their representation relations: the Inferential

28§6.2 for the Inferential Conception, and §4.2.2 and §7.2.1 for the PMA.

87

Conception of Bueno, Colyvan and French, and Pincock’s Mapping Account.

In the next chapter I will outline the accounts and explain how they can be

considered accounts of faithful epistemic representation.

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4 | Structural Relation Accounts

of Representation

4.1 Introduction

In the previous chapter, I argued that scientific representation should be un-

derstood to be a subspecies of epistemic representation, and suggested that

it is a subspecies of faithful epistemic representation. I accepted Contessa’s

arguments against Suárez, that surrogative inferences have to be grounded in

something, that we have to explain in virtue of what we are capable of using

a representing vehicle to draw surrogative inferences about a target. Con-

tessa argued that interpreting a representational vehicle in terms of a target

grounded the drawing of surrogative inferences, and that the ability to draw

valid surrogative inferences with a vehicle made that vehicle an epistemic rep-

resentation for that user. He also argued that a partially faithful representation

is an epistemic representation that allows for the drawing of some sound infer-

ences from the vehicle to the target. In this chapter I will be arguing for how

to understand two accounts of structural representation as providing accounts

of epistemic and faithful epistemic representation.

Contessa talks of requiring an account of faithful epistemic representation

in addition to an account of epistemic representation. We need an account

of faithful epistemic representation in order to explain how models represent

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phenomena, when those models contain various types of misrepresentations. It

is not clear whether Contessa is claiming that an account of faithful epistemic

representation is a separable account that provides only the faithful part of

such representation, or if it is capable of providing the epistemic part as well.

This is an issue which will be briefly explored in the course of the wider argu-

ment.

The structure of this chapter is as follows. I will outline the details of the

Inferential Conception (IC) and Pincock’s Mapping Account (abbreviated to

‘the PMA’ for ‘the Pincock Mapping Account’) in §4.2. I will then attempt

to fit these two accounts into Contessa’s approach. In §4.3.1 will argue that

while the IC cannot provide an account of epistemic representation, it can

provide an account of faithful epistemic representation. This is because the IC

only involves structural relations and the intentions of agents. The interpreta-

tion of the vehicle is provided by the structural relations, which also provide

the gradable notion of faithfulness. In §4.4.1 I will outline that the partial

structures framework used by the IC allows it to provide a measure on the

faithfulness of a representation, and promise to account for the relationship

between the faithfulness and soundness of a representation in Chapter 6. In

§4.3.2 I will argue that the PMA fulfils the criteria set out by Contessa for

analytic interpretations. In this section I also discuss how Pincock’s views on

representational inferences can be accommodated by Contessa’s approach. In

§4.4.2 I will argue that despite initial problems, the PMA can be considered an

account of faithful epistemic representation, providing the gradability of faith-

fulness through the use of various types of structural relations, in a manner

akin to that recommended by Contessa. I am only able to sketch how both

accounts can provide faithful epistemic representation in this chapter. A full

investigation is pursued in the subsequent chapters, where the question of how

faithfulness and usefulness are related is explored in detail. A choice between

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the accounts rests on the result of this investigation.

4.2 Structural Mapping Accounts of

Representation

4.2.1 The Inferential Conception of the Applicability of

Mathematics

The IC draws inspiration from Suárez’1 and Swoyer’s2 claims concerning sur-

rogative inferences, Hughes’ DDI account of representation3 and the partial

structures programme.4 One can see these three influences playing key roles

in the account: representation occurs due to the existence of multiple (partial)

mappings between (partial) structures, for the purposes of facilitating surrog-

ative inferences. The account itself closely resembles the structure of Hughes’

DDI account, with three stages involved in the representation of a vehicle by

a target.5

In this section I’ll provide an outline of the partial structures framework

and how it is employed by the account at each of its three stages. As I am

concerned with the mechanics of representation, ontological considerations will

not be pursued. A defence of the majority of the ideas behind this account has

already been presented in response to the various arguments against structural

1Suárez (2003, 2004)2Swoyer (1991)3Hughes (1997)4Bueno (1997, 1999), Bueno et al. (2002), da Costa & French (2003), French (1999,

2003), French & Ladyman (1998) amongst others.5The account is initially presented as being a way of understanding how mathematics

can be applied to the world in Bueno and Colyvan’s (2011). It was quickly developed toaccommodate wider issues involved in mathematical representation and scientific representa-tion more generally by Bueno and French in their (2011) and Bueno & French (2012). Thesepapers also helped to locate the IC within the partial structure version of the Semantic Viewadvocated by Bueno and French. This development does not alter the core principles of theaccount, so where the papers talk of an “application” of mathematics, “applying” or “applied”mathematics, I will talk of representation via mathematics, a mathematical representationor (representational) vehicle.

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accounts in the previous chapter (see §3.1), though it might be useful to re-

member that the IC is not a purely structural account. Explicit (and frequent)

use is made of its allowance for “additional pragmatic and context-dependent

features in the process of applying mathematics” (Bueno & Colyvan 2011, pg.

352).6

The Partial Structures Framework7

A partial structure is typically represented as a set-theoretic construct, A =

⟨D,Ri⟩i∈I , where D is a non-empty set, Ri is a partial relation and i ∈ I is

an appropriate index set. A partial relation Ri is a relation over D which is

not necessarily defined for all n-tuples of elements of D. We can understand

each partial relation R as an ordered triple, ⟨R1,R2,R3⟩. R1, R2 and R3 are

mutually disjoint sets, with R1 ∪ R2 ∪ R3 = Dn, such that: R1 is the set of

n-tuples that belong to R, R2 the set of n-tuples that do not belong to R, and

R3 is the set of n-tuples for which it is not defined whether or not they belong

to R. When R3 is empty, R is a normal n-place relation that can be identified

with R1.

For partial truth, we can understand a partial structure as the set-theoretic

construct: X = ⟨Z,Rj,P ⟩j∈J , where Z is a non-empty set, Rj is a partial

relation, and j ∈ J is an appropriate index set. P is a set of sentences of

a language L which is interpreted in X .8 As before, for some j, Rj might be

empty (as it was for i = 3, R3, above). P may also be empty.

The set that the partial relations are defined over, D and Z respectively,

denote the set of individuals in the domain of knowledge9 modelled or repre-

6Bueno and French go through the arguments presented by Suárez and defend the ICagainst them in §9 of their (2011).

7For this section I will reproduce the introduction to partial structures from Bueno et al.(2002) and da Costa & French (2003).

8As we are dealing with partial structures in terms of truth, we have to employ modeltheoretic machinery. See Hodges (1997) for an introduction to model theory.

9With respect to the domain of knowledge, ∆, rather than understanding D as simplydenoting the set of individuals, we might also understand this in terms of of a ‘data structure’,

92

sented in the particular case at hand, and the family of partial relations, Ri or

Rj, model or represent the various relationships that hold between the indi-

viduals. P can then be regarded as a set of distinguished sentences of L, which

might include, for example, observation statements concerning the domain.

A partial structure, A, can be extended into a total structure B (or X to

Y). This total structure can be described as an A-normal structure, where

B = ⟨D ′,R′i⟩i∈I , if:

(i) D = D ′;

(ii) every constant of the language is question in interpreted by the same

object both in A and in B; and

(iii) R′i extends the corresponding relation Ri, in the sense that each R′

i is

defined for every n-tuple of objects of its domain.

For the partial structures used for partial truth, there is a fourth condition:

(iv) If S ∈ P , then B ⊧ S

where S is a sentence in L. This can be understood more loosely as follows: a

total structure Y is called X -normal if it has the same similarity type as X , its

relations extend the corresponding partial relations of X , and the sentences P

are true (in the Tarskian sense, as the notion of partial truth is supposed to be

an extension of Tarskian truth) in Y . This allows us to say that S is partially

true in X , or in the domain that X partially modelled or represents, if there

is an interpretation I of L in an X -normal structure Y and S is true in the

Tarskian sense in Y . This can also be phrased as the claim that S is partially

true in the structure X if there exists an X -normal Y in which S is true in the

correspondence sense. If S is not partially true in X according to Y , then S

is said to be partially false in X according to Y . A major advantage of this

as in (da Costa & French 2003, pg. 17).

93

notion of partial truth is that it “captures the gist of the idea of a proposition

being such that everything occurs in a given domain as if it were true (in the

correspondence sense of truth)” (da Costa & French 2003, pg. 19).

The notion of partial structures also includes the notions of partial iso-

morphisms and homomorphisms. Let A = ⟨D,Ri⟩i∈I and B = ⟨E ,R′i⟩i∈I be two

partial structures (with Ri = ⟨Ri1,Ri2,Ri3⟩ and R′i = ⟨R′

i1,R′i2,R

′i3⟩ as above).

We can define a function f from D to E as a partial isomorphism between A

and B if:

1 f is bijective; and

2 for x and y in D:

(2.i) Rk1xy → R′k1f(x)f(y); and

(2.ii) Rk2xy → R′k2f(x)f(y).

If Rk3 and R′k3 are empty, i.e. we no longer have partial structures but ‘total’

structures, we recover the standard notion of isomorphism.

We can define f from D to E as a partial homomorphism between A and

B if for every x and every y in D:

(1) Rk1xy → R′k1f(x)f(y); and

(2) Rk2xy → R′k2f(x)f(y)

Again, if Rk3 and R′k3 are empty we recover the standard notion of homomor-

phism.

Outline of the Inferential Conception

The IC is predicated upon the claim that “the fundamental role of applied [i.e.

representational] mathematics is inferential” (Bueno & Colyvan 2011, pg. 352):

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by embedding certain features of the empirical world into a mathematical

structure, it is possible to obtain inferences that would otherwise be

extraordinarily hard (if not impossible) to obtain.

In order to accommodate this inferential role, the proponents of the IC claim

that establishing mappings between the target system and the mathematical

structure is “crucial”. One of the problems with such a claim that is frequently

raised is that the target system does not possess a structure, in that it is not a

set-theoretic object, nor a mathematical object.10 The solution to this problem

is to adopt the attitude that the world has an assumed structure: “that there is

some natural structure of the [target system] or that an appropriate structure

can be imposed upon the [target system]”, and that this is not a trivial matter,

nor that there is a unique structure (Bueno & Colyvan 2011, pg. 353, endnote

17).11

Adopting the position that there is an assumed structure in the target sys-

tem is only part of what Bueno and Colyvan mean by “empirical set up”. They

claim that the IC is idealised in two respects, one of which is that there is a

“sharp distinction” between the empirical set up and the mathematical struc-

tures.12 This is because “very often the only description of the set up available

will invoke a great deal of mathematics”, which leads to the situation where

it is “hard to even talk about the empirical set up in question without leaning

heavily on the mathematical structure, prior to the immersion step” (Bueno

& Colyvan 2011, pg. 354). One should therefore understand the empirical

set up to be “the relevant bits of the empirical world, not a mathematics-free

description of it”.

The core of the IC is a three step scheme, which can be understood as an

10Bueno and Colyvan recognise this problem, noting that “the world does not comeequipped with a set of objects . . . and sets of relations on those” objects (Bueno & Colyvan2011, pg. 347).

11For a criticism of the SMA along the lines of non-unique structures, see Baker (2012).12The second is dealt with below.

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Empirical Set Up

Immersion 1 Immersion 2

Model 1 Model 2

Interpretation 2Interpretation 1

Derivation

Figure 4.1: A schematic diagram representing the three steps of the Inferential Con-ception, with the immersion and interpretation steps iterated.

extension to Hughes’ DDI account (Bueno & Colyvan 2011, pg. 353-354):

Immersion: establish a mapping from the empirical set up to a con-

venient mathematical structure. This step aims to relate the relevant

aspects of the target system to the appropriate mathematical context.

The choice of mapping is a contextual matter, dependent upon the par-

ticular details of the representation.

Derivation: consequences are drawn from the mathematical formalism,

using the structure obtained in the immersion step. Bueno and Colyvan

consider this the key point of the application process.

Interpretation: the mathematical consequences (obtained in the deriva-

tion step) are ‘interpreted’ in terms of the initial empirical set up. Here,

an interpretation is understood to be a mapping from the mathematical

structure to the initial empirical set up. The mapping does not need

to be the inverse of the one used in the immersion step. No problems

emerge so long as the mappings are defined for suitable domains.

The immersion step does not have to be immediately followed by the deriva-

tion step. One can iterate the immersion mapping, to embed the first math-

96

ematical structure into another (Bueno & French 2012, pg. 91-92).13 The

immersion mapping is what gives representations their content on the IC. As I

will explain, this content is uninterpreted mathematics. The content of repre-

sentations is what constitutes the models manipulated in the derivation step.

The general idea behind structural relation accounts of representation is that

the content of representations is itself structural. Thus the models are under-

stood in terms of their structural properties.

The derivation step initially seems simple: one performs some mathemat-

ical manipulations on the structured obtained from the immersion step and

obtains a result. I will refer to this result as the ‘resultant structure’. It

is important to get clear about what is being manipulated in this step. In

mapping from the initial empirical set up to the model used in the derivation

step, via the immersion mapping, we shift from interpreted mathematics in the

empirical set up to uninterpreted mathematics in the derivation model. The

derivation step therefore involves the manipulation of an uninterpreted math-

ematical structure and so the resultant structure is uninterpreted. This is why

the interpretation step is required: we have to map the resultant structure

to an empirical set up which provides physical interpretation for the mathe-

matics.14 However, there might be a problem for this understanding of the

IC when one takes into account the second respect in which the IC is ide-

alised. This is the realisation that “the mathematical formalism often comes

accompanied by certain physical ‘interpretations” ’ (Bueno & Colyvan 2011,

pg. 354). This threatens both the definition of interpretation as a mapping

from a mathematical structure to an empirical set up, and the notion that the

13As can be seen in Fig 4.1, this requires the interpretation mapping to be iterated aswell. This is because the first model is mapped to the second model by the second immersionmapping. The resultant structure will therefore be ‘of’ the second model. In order to gainany physical insights from this resultant structure, we have to interpret in terms of the firstmodel.

14A problem arises if one does not provide such a mapping: “without an inverse mappingthe mathematics remains uninterpreted and says nothing about the empirical system it issupposed to be representing” (Bueno & Colyvan 2011, pg. 349, endnote 8).

97

content of a representation on the IC is a pure mathematical structure. I will

return to this possible problem in Chapter 6 when I outline what I call the

epistemic problem for the IC.15 For now, I propose that these ‘interpretations’

are not strictly interpretations, but rather heuristic, pragmatic and contextual

considerations that will help guide the choice of manipulations that will be

performed on the mathematical structure obtained from the immersion step.

The interpretation step is straight forward. However when discussing how

the IC is able to provide a framework within which one can conceptualise the

selection of appropriate mathematical structures, Bueno and Colyvan claim

that “we need not think of the immersion step as being logically prior to the

interpretation step” (Bueno & Colyvan 2011, pg. 356-357). This appears to

be because the selection of an appropriate mathematical structure arises from

going back and forth between the immersion and interpretation steps. This

claim threatens the idea that the content of representations on the IC is a pure

mathematical structure. The threat comes from the possibility that we end

up with interpreted mathematics in the derivation step due to the interpre-

tation step coming before the immersion step. In response to this, I propose

that for the initial application of mathematics to a target system, the first

time such a system is represented with mathematics, the immersion step must

come first. This is because there is no mathematical structure to map to the

target system via an interpretation mapping. Once the initial immersion and

interpretation mappings have been performed, a chain of iterations of the three

steps is established, leading to the situation where any non-initial immersion

or interpretation mapping is not logically prior to any other (granting that the

structures involved allow this). The reason for adopting this line of argument

is that I see maintaining a pure, uninterpreted, mathematical structure as the

content of representation on the IC as vital to its success when addressing

15See §6.2.2, page 168.

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idealisations, as well as the only viable response to the epistemic problem.

Finally, I turn to the how he IC accounts for the relationship between the

agent and the representational vehicle and target. The challenge is to identify

what the intention relates in a non-problematic way. Does the intention relate

the agent to the vehicle, to the target, to both, or to the structural relation

itself? Is the intention part of the representation relation? One potential

problem that can arise from including the intention that a vehicle represents

a particular target in the representation relation is that this may prevent the

vehicle from representing other targets (Bueno & French 2011, pg. 886-887).

The idea is that by including the intention of the scientist that a vehicle repre-

sents a particular target would fix that target as the only target for the vehicle.

For example, if we privileged the intention of Einstein that special relativity

represents the behaviour of rods and clocks, then Minkowski would not have

been able to use the theory to represent four dimensional spacetime. Bueno

and French argue that in order to avoid such a situation not only should we

not include the intention of agents in the representation relation, but that “we

must allow for pragmatic or broadly contextual factors to play a role in select-

ing which of these relationships to focus on”, and that the intentions should

be separated from the underlying mechanism (i.e. the partial morphisms) so

that the original representing agent’s intentions can be “overridden” by other

agents who wish to use the vehicle to represent some other target.

The approach favoured by Bueno and French is for the intention to pick

out the structural relation between the target and vehicle. i.e. to pick out the

relevant partial morphisms responsible for the immersion and interpretation

steps. One might worry that this approach to intentions does not solve the

non-sufficiency argument. After all, merely pointing to an isomorphism was

the cause of the insufficiency claim. The response to this challenge is to high-

light the role the intentions of the agent, along with contextual and pragmatic

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factors, play in identifying the structure of the final empirical set up.

On the IC multiple interpretations of a resultant structure correspond to

different partial mappings from that resultant structure to different partial

structures in the empirical set up. The intentions of the scientist and other

pragmatic and contextual factors will pick out a particular partial morphism

and therefore a particular empirical set up, i.e. a particular target. Remember

that an empirical set up is a particular part of the world, suitably described

in terms of interpreted structures. The interpretation associated with the

empirical set up, along with contextual factors such as what use the agent is

putting the representation to, pick out that empirical set up from the other

structures that the resultant structure is partially -morphic to. Thus on the

IC the representational relation consists in the immersion and interpretation

mappings, and the intentions of the agents relate the agents to these mappings.

Both these mappings and the intentions are necessary for a representation to

occur, along with the pragmatic and contextual considerations.

This concludes the introduction of the IC. I now turn to outlining Pincock’s

Mapping Account.

4.2.2 Pincock’s Mapping Account

In his (2012), Pincock is interested in establishing what the contribution of

mathematics is to scientific representations, how it can be involved in the

confirmation of scientific claims, its role in scientific inferences, how it can

contribute to scientific explanation and what conclusions can be drawn for the

realist/anti-realist debate in light of the contributions he identifies. Because

of these wide ranging goals, Pincock’s account of mathematical representation

is not schematically set out, but rather outlined abstractly, then expanded

upon through numerous examples. This is both useful and problematic: it

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allows one to see how certain parts of his account reveal the contribution of

mathematics in the various areas he is interested in, yet it causes problems in

identifying a clear, concise statement of what his account consists of. In light

of this, I shall provide as clear an outline of his account as I can in this chapter.

Further details will be provided through applying the PMA to my case study

of the rainbow in Chapter 7.

Pincock takes the content of mathematical scientific representations to be

“exclusively” structural, in that the “conditions of correctness . . . [imposed]

on a system can be explained in terms of a formal network of relations that

obtains in the system along with a specification of which physical properties

are correlated with which parts of the mathematics” (Pincock 2012, pg. 25).

A representation is obtained by denoting a target and a structure to act as a

representational vehicle. A structural relation is then identified between the

target and the structure, and the “specification” relation is identified, which

helps inform the content of the vehicle, including some form of interpretation

of the mathematics in terms of the target. The exact content of the repre-

sentation will depend on the application at hand, in that various parts of the

mathematical structure might become “decoupled” from its interpretation and

what is determined to be ‘intrinsic’ or ‘extrinsic’ mathematics. For example,

some predictions are considered to be extrinsic mathematics and so are not

part of the content of the representation. Pincock’s account is very close to

agreeing with Contessa’s idea of an analytic interpretation.16 I will show that

this is the case in §4.3.2. In this section I will expand upon Pincock’s notions

of content, introduce the distinction between intrinsic and extrinsic mathe-

matics and the related notion of core concepts, and detail the structural and

specification relations.

16At one point Pincock comes close to setting his account out in the same way as Contessa(Pincock 2012, pg. 257).

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Pincock’s Notion of Content

Pincock distinguishes between theories, models and representations (Pincock

2012, pg. 25-26):

Theory A theory for some domain is a “collection of claims” that aim

“to describe the basic constituents of the domain and how they interact”.

Model Any entity that is used to represent a target system [i.e. a

representational vehicle]. We “use our knowledge of the model to draw

conclusions about the target system”. They may be concrete entities, or

mathematical structures.

Representation A model with content.

Pincock takes contents to “provide conditions under which the representation

is accurate”. Understood schematically, this agrees with the IC: offering a

mathematical scientific representation can be summarised as the claim that

“the concrete system S stands in the structural relationM to the mathematical

system S∗” (Pincock 2012, pg. 28). A representation is correct if “both systems

exist and the structural relation obtains”, otherwise it is incorrect.

Before exploring the various notions of content Pincock works with, a quick

comment should be made concerning two of the assumptions he works with.

The first assumption is that scientists are capable of referring to the world in

some weak sense of ‘refer’. That is, they are able to refer to “the properties,

quantities and relations that constitute their domain of investigation” (Pincock

2012, pg. 26). This weak sense of ‘refer’ is cashed out in a counter-factual way.

Pincock explicates this notion through the example of phlogiston, and thereby

addresses the mistargeting problem of non-existent targets:

In specifying the contents of the contents of [the representations of phlo-

giston], I take the scientist’s ability to talk about phlogiston for granted.

But . . . I do not want it to follow that the phlogiston representations

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were without content. Instead, I would describe the situation as one

where the scientists were referring to phlogiston in the minimal sense

that they were coherently discussing a substance that could have ex-

isted, and had it existed, some of their phlogiston representations would

have been correct. As it stands, such representations all turned out to

be incorrect.

The second assumption is mathematical Platonism, though this assumption

is later weakened to semantic realism about mathematics (Pincock 2012, §9).

Pincock argues that the anti-realist positions of Hellman and Lewis are consis-

tent with his account of mathematical representation and of the Indispensabil-

ity Argument. I will not engage with Pincock’s views on the Indispensability

Argument in this thesis. However, I think that the approach Pincock has

taken to reach these conclusions is the right one. He has investigated the way

in which mathematics represents the world and produced an account of rep-

resentation that has consequences for which mathematical ontologies we are

able to adopt. That is, his answer to the problems of representation and the

applied metaphysical question have set limits on the available ways we have

to answer the pure metaphysical question, as I argued they would in §2.3.1.

The importance of these two assumptions is that they grant the ability to

agents to refer to both mathematical and physical entities when attempting

to represent the world mathematically. This can be seen in two ways. First, if

(schematically) representations are correct due to the existence of a concrete

system S, a mathematical system, S∗, and a structural relation M holding

between them, then an agent performing a representation requires “referential

access” to the mathematical system S∗ in order to adopt a belief with the

structural content Sc, i.e. the content of the representation (Pincock 2012, pg.

28). Second, they allow Pincock to adopt semantic internalism for mathemat-

ical concepts and semantic externalism for physical concepts (Pincock 2012,

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pg. 26-27).17

Now for the various notions of content. Pincock distinguishes four types

of content, which roughly correlate with differing levels of sophistication of

representation, though he warns the reader that he does not “claim that every

representation goes through these stages [of differing contents] or that scientists

would always be happy making these distinctions”. Rather, one is to think of

these distinct types of content as a “tool to help explain how to find the content

in a given case” (Pincock 2012, pg. 26). The order in which they are introduced

correlates with this story of increasing sophistication in representation.

The first notion of content Pincock distinguishes is called basic content.

This is the type of content that arises from simple structural mappings (such

iso- and homo- morphisms) and the basic elements of the concrete and mathe-

matical systems (Pincock 2012, pg. 29). To get a grip on what Pincock means

by “basic elements” of concrete and mathematical systems, one can contrast

the examples he provides and his statements concerning those examples in his

§2.2 - Basic Contents and those he provides in his §2.3 - Enrich Contents in

particular the statements concerning derived elements. Derived elements are

a type of mathematical element Pincock distinguishes en route to his second

type of content, enriched contents.

In his §2.2 Pincock provides the example of concrete system composed of

a group of 5 people and the relation of ‘order of age’, which is held to be

isomorphic to the natural numbers 1 through 5 and the less than relation (on

the assumption no two individuals share the same age). In his §2.3, Pincock

provides the example of the heat equation, a partial differential equation:

α2uxx = ut (4.1)

17These positions are again altered slightly later when Pincock considers Wilson’s ap-proach to concepts and the revision of concepts. See Pincock (2012) Chapter 13, and Wilson(2006).

104

When applying this equation, one might expect the accuracy conditions to be

along the lines of requiring an isomorphism between the temperature at each

point in time and the set of ordered pairs (x, t) picked out by the solution to

(4.1) (Pincock 2012, pg. 30). But this would cause two problems. First, we

do not expect our representation to be as accurate as an isomorphism would

demand. Second, temperatures are not defined on a single spatial point, but

rather thought to be a property of spatial regions. One way of responding

to these problems is to stick with the basic contents and conclude that all

such representations are inaccurate. This response faces serious difficulties in

accounting for our use of our representations if they are nearly all taken to be

inaccurate. Pincock opts for a different approach, which I will outline after

establishing what constitutes basic elements.

From the above comments I propose that the basic elements of concrete

and mathematical systems are as follows. For concrete systems they are things

like space time points and properties at space time points. For mathematical

systems they are things like numbers, properties of numbers, and ordered pairs

of numbers. Thus anything more sophisticated than these elements will fall

under the banner of ‘derived’ elements. So things like functions, properties

over regions and so on.18 These derived elements might constitute all of the

elements involved in a representation, and are especially prevalent in idealisa-

tions (Pincock 2012, pg. 29). I gather that the content of any representation

that involves derived elements will not be exclusively basic. But this does not

mean that all representation that involve derived elements will have content

18It should be noted that Pincock appears to only introduced derived elements in termsof the mathematical structures: “these derived elements in the mathematics will be used torepresent physical entities beyond those [that] can be related directly to the mathematicalentities that appear in the domain of the mathematical structure” (Pincock 2012, pg. 29,my emphasis). However, given Pincock’s claims about temperature not being a fundamentalproperty of an iron bar (the system he applies the heat equation to in his example), I believeit makes sense to extend this notion to elements of the target domain. I take this to be whatPincock refers to as “derived quantities” in the introduction to his §2.4 - Schematic andGenuine Contents. Further, nothing much rests on this. The role of derived elements inidealisations will be far less significant than that played by the idealising assumptions.

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exclusively composed of enriched content; rather enriched, and the other two

types of content will include derived elements.

Enriched contents are another way of responding to the two problems for

the application of the heat equation. We move from basic contents to enriched

contents, employing derived elements in our concrete and mathematical sys-

tems. An enriched content can be understood to be a mathematical structure

where a representational intention and/or details of the derived elements of the

mathematical structure alter the structural relationM between S and S∗, both

by employing a more sophisticated relation than an iso- or homo- morphism,19

and by allowing the relation/specification relation to have an influence on the

content of the representation (Pincock 2012, pg. 31). Pincock explains which

part of the representation of heat diffusing through an iron bar is constituted

by enriched content (Pincock 2012, pg. 30-31):

In the heat equation, we have to work with small regions in addition

to the points (x, t) picked out by our function. We should take these

regions in the (x, t) plane to represent genuine features of the tempera-

ture changes in the iron bar. This representational option is open to us

even if the derivation and solution of the heat equation seem to make

reference to real-valued quantities and positions. We simply add to the

representation that we intend it to capture temperature changes at a

more coarse-grained level using regions of a certain size which are cen-

tered on the points picked out by our function. The threshold here

can be set using a variety of factors. These include our prior theory

of temperature, the steps in the derivation of the heat equation itself

or our contextually determined purposes in adopting this representation

to represent this particular iron bar. My approach is to incorporate all

of these various inputs into the specification of the enriched content.

19By ‘more sophisticated’ I am referring to Pincock’s use of employing mathematicalterms within the structural relations. I cover the introduction of these terms below, on page111.

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The enriched content has a much better chance of being accurate as it

will typically be specified in terms of the aspects of the mathematical

structure which can be more realistically interpreted in terms of genuine

features of the target system.

Enriched contents are inadequate for accounting for all of the features of

representing the diffusion of heat through an iron bar (and all of the features

of a large number of idealisations). Specifically, the practice of representing

the iron bar as being infinitely long. This practice requires a different notion

of content than can be supplied by enriched contents, as such a representation

seems to be doing something different than claiming that the iron bar is approx-

imately infinitely long, or that it is infinitely long given some error term. How

one interprets these idealisations threatens to commit one (or certain realists

at least) to inconsistent properties.20 For example, when employing the heat

equation we make the idealisation that the bar is infinitely long, yet we also

know that the bar is only 1m long from our observations. Thus by adopting

the heat equation representation, this representation in conjunction with our

observations attribute inconsistent properties to the bar. Pincock’s approach

to these sorts of idealisations is motived by a rejection of these inconsisten-

cies. The idea is that such representational practices “decouple” or “detach”

the relevant part of the mathematical structure from its “apparent physical

interpretation” (Pincock 2012, pg. 32). Thus, when one sets the length of the

iron bar to infinity, one is not claiming that the bar is infinitely long. Rather,

one detaches the mathematics from its “prior association with a physical quan-

tity in the physical system” and so “these structural similarities between the

mathematical system and the physical system become irrelevant to the cor-

rectness of the overall scientific representation.” Most importantly, “it is not20See Maddy (1992) and Colyvan (2008) for arguments that such idealised properties are

indispensable to science and so commit us to inconsistent claims when they are employed.Pincock’s rejection this analysis of these idealisations will be covered in §7.2.1. He gives hisresponse in §5 of his (2012).

107

part of the content of the scientific representation that the bar is infinitely long

or, perhaps, that it is any length at all ” (my emphasis). In doing so, one shifts

to schematic contents. One way of understanding schematic contents is that

they are pieces of the mathematical structure which have no interpretation.

Alternatively, due to the previous physical interpretation this content had we

are able to track that the representation says nothing about a particular prop-

erty of the system - in this case the length of the iron bar.

Schematic contents can be understood to be schematic in two senses. First,

schematic contents are “impoverished” compared to enriched contents, in that

they place less restrictions on the way the system has to be for the representa-

tion to be correct. Second, the schematic content, often a particular variable

in the mathematical structure, is considered to be an “unspecified parameter”

as it now has no physical interpretation. These parameters reflect the fact

that the representation provides no information on the property they were

previously interpreted as. This should be contrasted with a situation where a

representation claims the iron bar has no length. I presume this would amount

to setting the length variable to 0.

By attributing values to unspecified parameters, one moves from schematic

contents to genuine contents (Pincock 2012, pg. 32). How the move from

schematic contents to genuine contents occurs depends on various contextual

considerations, such as the differences in the target systems, the purposes of

the investigation and so on. However, these goals do not get included in the

content of the representation, because “it is no part of the accuracy conditions

of the representation that the representation serve the goals and purposes of

the scientist” (Pincock 2012, pg. 33). I take this to mean that the specification

is guided by the goals of scientists, but that it does not include these goals.21

21The above exposition can only serve as an introduction to the notions of content thatPincock works with. For the clearest, although rather simple, examples of how Pincock’snotions of content can be applied to actual representations, see his example of how torepresenting traffic flow (both in dynamic and steady states) and the example of the bridges

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Core Concepts and Internal & External Mathematics

The different notions of content are not the whole story of what mathematics

constitutes the content of representations. The notion of core concepts and

the distinction between intrinsic and extrinsic mathematics play a role. While

both of these ideas are influenced by Pincock’s views on the inferential role of

mathematics in scientific representations (which shall be addressed in §4.4.2),

they can be introduced independently of those considerations.

Pincock distinguishes the intrinsic and extrinsic mathematics as follows

(Pincock 2012, pg. 37):

Intrinsic Mathematics “Mathematics that is used in directly speci-

fying how a system has to be for the representation to be correct”; the

mathematics that “appears in the mathematical structure involved in the

content”.

Extrinsic Mathematics Mathematics that is not intrinsic, or provides

other contributions using other mathematics to the intrinsic mathemat-

ics. e.g. The “mathematics used to derive and apply a set of equations

might be different to the mathematics of the equations themselves”.

What might be considered an odd consequence of this distinction is that par-

ticular predictions about a target system are not part of the content of a

representation. Pincock explicitly states that “extrinsic mathematics [can be]

used to derive a prediction from a mathematical scientific representation”, the

core example of this being the solving the partial differential equations involved

in the heat equation, (4.1) (Pincock 2012, pg. 38).

Establishing what is the intrinsic mathematics, and therefore the content,

of a representation is not always straight forward. For example, one might be

representing a section of an iron bar via a series of real numbers. This raises

of Könngisberg (Pincock 2012, pg. 48-58).

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questions of the following kind. Are all real numbers part of the intrinsic math-

ematics? What about complex numbers, which are necessarily linked to the

real numbers? Pincock employs the notion of ‘core conception’ to help settle

this issue. In this case, complex numbers are not taken to be part of the core

conception of real numbers, and so would constitute extrinsic mathematics.

But how does one establish what a core conception is?

The main motivation for Pincock’s adoption of the notion of core concep-

tions is a problem related to inference. For now, this can be summarised as

an attempt to find a middle path between the two extreme responses to the

question of whether metaphysical necessity is respected by logical inference.

These two extremes being the claim that either all metaphysical necessities

are respected, or none of them are (Pincock 2012, pg. 35). The idea is that

there is some ‘core’ elements or features of mathematical terms that must be

grasped in order to be said to have an understanding of the terms. For ex-

ample, transitivity is a part of the core conception of ‘greater than’. Thus,

when one considers the mathematical techniques required to solve the partial

differential equations involved in the heat equation, one should see that these

techniques “go far beyond what could be reasonably required to understand the

equations themselves”. As part of explaining what constitutes “understanding”

here, Pincock appeals to the restricted faculties agents possess. That is, there

are limitations on the number of calculations, inferences, or steps of a deriva-

tion an agent can perform without the use of tools (such as pen and paper,

or a computer), and that these limitations play a role in delineating the core

conceptions. This can be summarised as the stipulation “that only a certain

level of complexity in [a] derivation can be tolerated”, where complexity is

judged by, for example, the length (i.e. the number of steps) of the derivation

(Pincock 2012, pg. 37).

Now that I have introduced the notions of content and core conceptions,

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and the intrinsic/extrinsic mathematics distinction, I will turn to explaining

Pincock’s position on what the structural and specification relations are, and

their role in Pincock’s account.

Pincock’s Structural Relations and Specification Relation

Pincock defines a structural relation as follows (Pincock 2012, pg. 27):

A structural relation is one that obtains between systems S1 and S2

solely in virtue of the formal network of the relations that obtains be-

tween the constituents of S1 and the formal network of the relations

that obtains between the constituents of S2 [where a] formal network

is a network that can be correctly described without mentioning the

specific relations which make us the network.

This is a fairly general notion and Pincock recognises that it must be tightened

up in terms of explicating what the range of acceptable structural relations are

in order for his account to succeed. Pincock is willing to accept the usual set

theoretic mapping relations of isomorphisms and homomorphisms (and pre-

sumably other such mappings). However, when introducing enriched contents,

Pincock argued that iso- and homo- morphisms are insufficient. They are re-

stricted to relating the basic elements of mathematical and physical systems

and so caused problems, prompting the shift to enriched contents. This shift

resulted in an alteration of the structural relationM between the two systems,

S and S∗. What this alteration consists in can now be explained.

Rather than be restricted to (simple) set theoretic structural mappings,

Pincock is open to incorporating mathematical notions in the structural re-

lations: “[he will] allow more intricate sorts of structural relations, including

those whose specification requires mathematics” (Pincock 2012, pg. 27). How

these structural relations are formed will depend on the representation at hand.

For example, when attempting to represent the temperature diffusion through

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an iron bar with the heat equation, (4.1), we must take into consideration the

fact that temperature is defined over regions, rather than points, of space. One

way of doing this proposed by Pincock is to consider an isomorphism between

the temperature at times, and u in the mathematical structure, subject to a

spatial error term ε = 1mm. The idea is that the structural relation includes

the ‘threshold’ of the spatial regions over which the temperature is defined.

The specification relation is first introduced as Pincock attempts to sharpen

up the notion of structural content (Pincock 2012, pg. 25, my emphasis):

. . . the content of mathematical scientific representations . . . is exclu-

sively structural [which means] that the conditions of correctness that

such representations impose on a system can be explained in terms of

a formal network of relations that obtains in the system along with a

specification of which physical properties are correlated with which parts

of the mathematics.

This initial statement is then refined by comparison to Suárez (2010), when

Pincock explains how he bridges the apparent gap between concrete systems

and the set theoretic descriptions of structures22 (Pincock 2012, pg. 29, my

emphasis):

Suppose we have a concrete system along with a specification of the rele-

vant physical properties. This specification fixes an associated structure.

Following Suárez, we can say that the system instantiates that struc-

ture, relative to that specification, and allow that structural relations

are preserved by this instantiation relation (Suárez 2010, pg. 9). This

allows us to say that a structural relation obtains between a concrete

system and an abstract structure.

I take these quotations to detail the two roles the specification plays:

22Pincock appeals to the distinction between systems and structures in Shapiro (1997).

112

i) Provide an interpretation of the mathematical structure.

ii) Fix the mathematical structure.

I also take these quotations to imply that the representation relation for Pin-

cock is at least a two-part compound relation, composed of a structural relation

and the specification relation. The agent is also involved in the representation,

but exactly how is not clear. The agent might be related to either the vehicle,

target and specification relation, or to the vehicle and target by virtue of the

specification relation, or in some other way. This will be clarified in §4.3.2,

when inference is discussed in more detail.

We can see how the specification plays its two roles by referring to how

enriched and schematic contents are identified. Enriched contents involve the

adoption of derived elements in the mathematical structure. Here, the spec-

ification can play the ii) role: we recognise that temperature is defined over

a region of space, and so must alter our mathematical structure to accommo-

date this feature of the world in order to have a possibly correct representation.

Pincock explains this in terms of “incorporating” the contextual features that

influence the choice of ε into the specification (Pincock 2012, pg. 31). Schematic

contents involve the decoupling of the (prior) interpretation of mathematical

variables. Here the specification will play the i) role, by simply not relating

the variable x to the length of the iron bar any more.

It appears that the specification relation is more than simply intentions.

Pincock states that the intention for the heat diffusion representation to cap-

ture regions of space is only one of the inputs to be incorporated into the

specification. He also lists such things as a theory of temperature and the

derivation of the heat equation as “inputs” to the threshold of regions to be

incorporated into the specification.

This concludes my introduction to the Inferential Conception and Pincock’s

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Mapping Account. I will now turn to establishing how one might understand

these accounts in terms of Epistemic Representation.

4.3 The Structural Mapping Accounts &

Epistemic Representation

In this section and the next I will be exploring how the IC and PMA can be

understood in terms of Contessa’s approach. prima facíe Contessa requires

separate accounts for epistemic representation and faithful epistemic repre-

sentation. He offers his Interpretational Account of epistemic representation

separately from his arguments in favour of a structural account of faithful

epistemic representation (Contessa 2011, pg. 126-130). The Interpretational

Account of epistemic representation consists in (what I have called) a global

denotation relation between vehicle and target, and Contessa’s analytic ac-

count of interpretation. While Contessa does not offer an account of faithful

epistemic representation, he indicates that he would advocate an account that

makes use of various types of structural relations in order to cash out the grad-

ability of faithfulness in terms of similarity of structure: “the more structurally

similar the vehicle and target are . . . the more faithful an epistemic represen-

tation of the target the vehicle is” (Contessa 2011, pg. 129-130). However, it

does not appear necessary for one to supply separate accounts, and cannot be

necessary if the IC and PMA are viable accounts of faithful epistemic repre-

sentation. Rather the requirement seems to be that there are three separable

parts of an account:

i) Global denotation: a relation that points the vehicle to the target.

ii) Interpretation of the vehicle in terms of the target: loosely characterised,

this requires the agent to “take facts about the vehicle to stand for (pu-

tative) facts about the target” (Contessa 2007, pg. 57).

114

iii) Gradable faithfulness relation: some relation that provides a suitable

answer to the question “in virtue of what does [a] model represent a

certain system faithfully” and is a gradable relation (Contessa 2007, pg.

68), (Contessa 2011, pg. 129).

The first two requirements constitute Contessa’s account of epistemic represen-

tation, while the third constitutes his (putative) account of faithful epistemic

representation. Whether the first two requirements are genuinely distinct re-

quirements is debatable. In §3.4.2 I highlighted that Contessa includes the

global denotation in his account of analytic interpretation in his (2007), but

separates it out in his (2011). All that is required is that the vehicle is aimed at

the target. This could be through explicitly distinct denotation and interpreta-

tion relations, through an interpretation relation that includes the denotation

relation, or some other compound interpretation relation.

To establish whether the IC and PMA satisfy these requirements and fit

into Contessa’s approach, I will ask the following questions of the accounts:

(1) Which part of the account provides the epistemic representation?

(2) How does the account establish faithful epistemic representation? in-

cluding:

(2.a) How does it supply a gradable notion of faithfulness?

(2.b) How does it account for the relationship between faithfulness and

usefulness?

Question (2.b) is motivated by Contessa’s arguments that faithfulness and use-

fulness do not always coincide (Contessa 2011, pg. 130-131). He argues for the

splitting of faithfulness and usefulness in response to Galilean idealisation, but

this notion can be generalised to any case of misrepresentation. What is sig-

nificant about this realisation is that what determines a useful representation

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is the purposes and the goals of the agent using the representation. A viable

account of faithful epistemic representation has to account for how less faithful

representations can be more useful than more faithful representations.

As part of answering these questions I will be concerned with the idea that

the relations responsible for epistemic representation have to be completely

separable from the relations responsible for faithful epistemic representation.

I take this to be a fair reading of the distinction Contessa argues for, rather

than a requirement that there are two separate accounts. Given the intro-

duction of the accounts above, it should be clear that the PMA can provide

separate relations that would account for epistemic representation and faith-

ful epistemic representation. It does not seem possible to provide separate

relations on the IC, due to the interpretation of the vehicle being provided by

the interpretation mapping, a structural relation. I will argue below that the

IC is evidence that Contessa is mistaken to think that epistemic and faithful

representation should always be separable, both in terms of distinct accounts

and in terms of there being distinct relations. This has no effect on adopt-

ing his approach and considering the IC and the PMA as accounts of faithful

epistemic representation.

I will now turn to answering (1). I will then provide outlines of answers

to question (2) in §4.4. Fully worked out answers to questions (2.a) and (2.b)

will be provided in Chapters 6 for the IC and 7 for the PMA.

4.3.1 The Inferential Conception

Representation occurs on the IC when an agent adopts a partial mapping

from some empirical set up to a mathematical model (the immersion step),

performs some mathematical manipulations to obtain a resultant structure

(the derivation step), and then interprets this resultant structure in terms of

the empirical set up by adopting another partial mapping, from the resultant

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structure to the empirical set up (the interpretation step). As explained above,

the target of the representation is picked out by the intention of the agent,

and the contextual and pragmatic considerations such as the use the agent is

intending to put the representation to. The intention picks out the immersion

and interpretation mappings. We can therefore point to the intention of the

agent and the pragmatic and contextual considerations as playing the role of

the global denotation relation, and the interpretation mapping as providing

the interpretation of the vehicle in terms of the target. These are the relations

that constitute the epistemic representation part of the IC.

There are two problems with this analysis of the IC. First, the way Con-

tessa set out his accounts of epistemic and faithful epistemic representation

implied that the relations that constitute the epistemic representation relation

should be distinct from those that constitute the faithful epistemic representa-

tion relation (whether these relations constitute two distinct accounts or not).

This is clearly not the case for the IC, as the relation which provides the inter-

pretation of the vehicle in terms of the target is a structural relation, which is

supposed to responsible for the faithfulness of the representation. Second, we

are only supposed to be able to draw inferences about our target after we have

adopted an interpretation of the vehicle in terms of the target. However, it ap-

pears that we can draw inferences before we adopt such an interpretation: the

derivation step appears to involve drawing surrogative inferences, and this step

finishes before we have interpreted the vehicle in terms of the target (before

the interpretation mapping is identified).

The first problem puts pressure on accepting the IC as an account of faith-

ful epistemic representation. I argued above that although there need not be

separate accounts of epistemic and faithful epistemic representation, Contessa

could be interpreted as requiring that there be relations that separately pro-

vide epistemic and faithful epistemic representation. The IC clearly cannot

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provide these relations, which leaves us with the choice of either rejecting the

IC as an account of faithful epistemic representation (as it does not include

relations for epistemic representation), or rejecting this interpretation of Con-

tessa’s statements (and possibly the statements that were interpreted).

There is strong evidence that we should accept the IC as an account of faith-

ful epistemic representation: the authors discuss the role it plays in providing a

framework for accommodating surrogative inferences; it is inspired by Hughes’

DDI account, which is an account of representation Contessa advocates as a

possible account of epistemic representation; and it introduces the structural

relations required to accommodate the faithfulness of representations.

I think that the correct course of action is to reject Contessa’s statements

concerning the distinctions between epistemic and faithful epistemic repre-

sentation as endorsing any kind of necessary distinction of either accounts or

relations. We can reject these statements while remaining consistent with Con-

tessa’s approach. Faithful epistemic representation is held to occur due to the

existence of a structural relation. If this relation does not exist, it need not be

the case that we have an epistemic representation, as this can be established

through different relations. We can have accounts of faithful epistemic rep-

resentation that use structural relations (and agents intentions) to establish

the interpretation of the vehicle in terms of the target as well as providing the

gradable notion of faithfulness.

A possible objection to this solution to the first problem is that we would

be unable to have situations where we thought we had a (partially) faithful

epistemic representation that then turned out to be a wholly unfaithful repre-

sentation. Contessa thinks that such cases exist (Contessa 2011, pg. 63). We

would be unable to have these situations on the IC because either there exists

a structural relation between the target and the vehicle, and so the vehicle is a

(partially) faithful epistemic representation, or the relation does not exist and

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so it is not a representation of any kind. If the structural relation does not

exist, there is no relation that can play the role of the interpretation relation,

and so the vehicle could not be an epistemic representation of the target.

I can provide two responses to this objection. Either we were always mis-

taken that the vehicle was a (partially) faithful representation, and so we

should not have been employing the IC anyway; some other account would

have to be provided to account for our taking of the vehicle to be a represen-

tation (e.g. Contessa’s Interpretational Account of epistemic representation).

Or the representation is still partially faithful, it is just not faithful in any way

we now deem useful. I have in mind here the response I gave in §3.3.1 and

§3.5 to the non-existent targets type of misrepresentation. If we ever thought

that the representation was partially faithful, then there must have been some

inferences that were accepted as sound at the time. Most likely, these would

be measurement results or predictions of measurement results. Provided we

leverage the contextual dependence of ‘use’, we will still be able to accept

these inferences as sound (i.e. rather than saying the model is providing both

measurement predictions and ontological statements, it is now only providing

a way of saving the phenomena.). There is still a structural relation between

the vehicle and the measurement results, even if we can now state it as being

less strong than initially thought.

The second problem can be resolved by recognising a distinction between

intra-vehicular inferences, and extra-vehicular inferences.23 The derivation

step involves inferences that are internal to the representing vehicle. As the

structure is an uninterpreted mathematical structure, we cannot take any in-

ferences we draw from it to be about anything but mathematics. This is the

entire reason for requiring the interpretation mapping. Thus simply drawing

consequences from the mathematical formalism is not a performance of surrog-

23The idea of intra-vehicular inferences I’m proposing here is similar to the notion thatmodels have an “internal dynamic” (Hughes 1997, pg. S331-S332).

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ative inference. To perform surrogative inferences, we have to infer something

from the vehicle to the target, that is, we have to interpret something about

the vehicle in terms of the target, which is what the interpretation mapping

provides. I therefore call the inferences drawn in the derivation step intra-

inferences: they are internal to the vehicle. Surrogative inferences are extra-

inferences, inferences that are external to the vehicle, they result in claims that

are external to the model. Thus, although we can perform inferences before

interpreting the mathematics, they are not the sort of inferences that Contessa

is referring to by surrogative inferences. On the IC, the interpretation mapping

provides the surrogative inferences. But again, which mapping we choose will

be dependent upon the pragmatic and contextual considerations. The IC will

involve a pragmatic and contextual ‘relation’ for each mapping involved in the

representation.

The above discussion shows that the IC and Contessa’s programme are

compatible. The broadly contextual and pragmatic considerations that are

involved in representation include such things as representational and contex-

tual intentions, and these considerations in conjunction with the immersion

and interpretation mappings constitute the part of the IC which allow us to

draw surrogative inferences with the representational vehicle. The gradability

part of faithful epistemic representation will be provided, as mentioned above,

by the partiality of the structures and mappings involved.

I now turn to outlining how the PMA deals with epistemic representation.

4.3.2 The Mapping Account

As indicated in §4.2.2, the PMA looks, prima facíe to be an account of mathe-

matical scientific representation that exemplifies Contessa’s notion of analytic

interpretation. This can be seen most clearly when Pincock defends his ap-

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proach from Fictionalist arguments. He summarises the process by which

representations obtain content as follows (Pincock 2012, pg. 257):

As I have presented things in this book a scientific representation gets its

content in three steps. First, we must fix the abstract structure which we

are calling the model. This is a purely mathematical entity. Then some

parts of this abstract structure must be assigned physical properties

or relations. At this second stage the parts of the purely mathematical

entity are assigned denotation or reference relations. Finally, a structural

relation must be given which indicates how the relevant parts map onto

the target system or target systems of the representation. At the end

of these three steps the representation has obtained its representational

content. This means that it is determinate how a target system has to

be for that representation be an accurate representation of that target

system.

In this quotation one can identify the two roles of the specification: fixing the

mathematical structure and providing an interpretation for the mathematical

structure.24 We can also see that the specification relation is distinct from the

structural relation.

I take Pincock’s account as conforming to Contessa’s notion of an analytic

interpretation. As such it does provide separate relations for epistemic and

faithful epistemic representation. In order to have an analytic interpretation,

the account needs to satisfy four conditions: taking the vehicle to denote the

target (the ‘global denotation relation’); taking every object in the vehicle

to denote one and only one object in the target; similarly for every relation

in the vehicle and target; and similarly for every function in the vehicle and

target. I take the specification relation to be capable of providing the ‘global

denotation relation’ as it includes representational intentions. These intentions24These roles were outlined in §4.2.2 - Pincock’s Structural Relations and Specification

Relation, on pg. 111.

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are involved in the specification’s first role, of fixing the structure. This is the

process of finding a mathematical structure that can be used to represent the

physical system. This is a little more involved than simply denoting a piece of

mathematics. The fixing of the structure does not just take into consideration

the representation intentions, but also features of the mathematical structure

that is being fixed (such as steps taken to derive the structure), features of the

theory or previous theories that are used to represent the phenomena, and so

on (Pincock 2012, pg. 31).

Further, this step only involves an uninterpreted mathematical structure

at this point. I consider it to be similar to searching for the correct tool.

The agent roots around their ‘mathematical tool box’ attempting to find the

right mathematical structure. The intentions inform the search, rather than

fixing any target. This reverses the direction of the intentions from how the

problem was posed above, where the intention was going from the vehicle to

the target. e.g. That E =mc2 would always have to represent rods and clocks

if Einstein’s intentions were included in the representation relation. Pincock

sets up the intention relations as going in the other direction: one has a target

and must find a structure to use as a representation. This way of looking at

the involvement of intentions, and how it avoids the problem, is reinforced by

the distinction drawn between the specification and the structural relation. i.e.

Intentions concern the identification of the vehicle, rather than the target or a

structural mapping. A consequence of this approach is that one might obtain

the structural mapping for ‘free’, due to choosing the structure of the vehicle.

This is not the only way in which the PMA uses intentions, however. When

outlining how the PMA will accommodate the gradable notion of faithfulness,

I will argue that the intentions will pick out different structural relations in

addition to their role here.

The other three conditions for an analytic interpretation can all be met by

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the second role of the specification, that of providing an interpretation. There

is evidence that the specification does this through denotation. In the above

quotation, Pincock describes the interpretation of the mathematical structure

to occur due to “the parts of the purely mathematical entity [being] assigned

denotation or reference relations” (Pincock 2012, pg. 257, my emphasis); when

discussing the heat equation, he talks of the physical and mathematical ba-

sic contents being isomorphic to each other, where the “set of ordered pairs

(x, t) picked out by the solution to our equation, where the first coordinate

denotes position and the second coordinate denotes time” (Pincock 2012, pg.

30). Further, I think it is possible to consider this denotation separate to the

isomorphism, both from how Pincock explains the situation here, and that the

previous quotation (from pg. 257) explicitly separates out both relations, talk-

ing of first the establishing the specification/interpretation relation and then

the structural relation. Thus I take the way the specification relation provides

an interpretation of the mathematical structure in terms of the target to be

an instance of Contessa’s analytic interpretation.

Inferences and Mathematical Necessity

There are two problems with the idea that Pincock’s account directly and sim-

ply fits into Contessa’s programme however. First, Pincock distinguishes his

approach to representation from inferential approaches, including Contessa’s.

Second, Pincock has a particular view on inference, which is motivated by

considering the relationship between the metaphysical necessities that mathe-

matics includes and logical consequence.

The first of these problems is not too significant. Pincock explains the

difference as (Pincock 2012, pg. 28):

Perhaps the main competitor to an approach based on accuracy condi-

tions tends to put inferential connections at the heart of their picture of

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representation. Inferential approaches must explain the scientific prac-

tice of evaluating representations in terms of their accuracy. While there

does not seem to be any barrier to doing this, I have found it more conve-

nient to start with the accuracy conditions. On my approach, inferential

claims about a given representation follow immediately from its accuracy

conditions: a valid inference is accuracy-preserving.

I take the distinction between Contessa’s and the IC’s, and Pincock’s ap-

proaches to be one of priority, which has no consequences for understand-

ing Pincock’s account in terms of epistemic representation. That is, Pincock

constructs his account starting from accuracy conditions, and addresses infer-

ences after he has made clear how representations provide accuracy conditions,

whereas the inferential approaches construct their accounts by starting with

a focus on how to guarantee inferences, and then establish how these lead to

accurate representations. In other words, we can understand Pincock to be

attempting to establish an account of faithful epistemic representation, which

entails an account of epistemic representation. Inferential approaches might

aim to establish an account of epistemic representation first and then extended

it to an account of faithful epistemic representation.25

The second problem is far more threatening. It originates in the approach

Pincock takes, focusing on accuracy conditions before inference. This results

in a focus on how scientific representations are confirmed to a greater extent

than the inferential approaches do.26 Pincock’s discussion of inference begins

with his adoption of the “prima facíe assumption that representations are

confirmed by entailing, perhaps in conjunction with auxiliary assumptions,25The advocates of the IC start from the position that mathematics is capable of providing

surrogative inferences. They construct the IC to show how mathematics can play thisrole. As such their starting position entails that the IC should be an account of epistemicrepresentation. However, as they adopt structural relations to account for the surrogativeinferences, they skip an account of epistemic representation and actually argue instead foran account of faithful epistemic representation.

26At least more than Contessa, who does not discuss confirmation, or the authors of theIC, who only discuss the consequences of establishing whether a representation is empiricallyadequate or not, rather than how the representations can be found to be confirmed or not.

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some predictions that are then experimentally verified. So we need to focus

on this inferential link between our representations and these predictions to

understand confirmation” (Pincock 2012, pg. 33, my emphasis).

Pincock approaches his analysis of the inferential link by comparing it to

our standard account of logical inference. He focuses in particular on the trans-

lation of English into logic, which allows us to make the “distinction between

logical and nonlogical terms . . . explicit”. Pincock goes on to argue, however,

that extending this approach to our mathematical scientific representations

“will distort their genuine inferential relations and obscure difficulties in con-

firming them”. To motivate this view, Pincock translates the sentences ‘the

number of fish is greater than the number of cats’, ‘the number of cats is

greater than the number of dogs’, and ‘so, the number of fish is greater than

the number of dogs’ into mathematical language:

a > b;

b > c;

∴ a > c

and logical language:

Nx(x is a fish) > Nx(x is a cat);

Nx(x is a cat) > Nx(x is a dog);

∴ Nx(x is a fish) > Nx(x is a dog)

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or:

G(ab);

G(bc);

∴ G(ac)

The problem with these two translations in to logical language is that there are

no logical terms; that is the greater than relation, >, has been translated into a

two-place predicate, which has no “fixed interpretation” (Pincock 2012, pg. 34).

As the two place predicate G(⋅⋅) does not have a fixed interpretation, when

analysed in terms of possible worlds there will be worlds where this argument

has true premises and a false conclusion. e.g. G(⋅⋅) could be interpreted as

‘greater than’, or ‘less than’. Thus, the above argument is invalid. For Pincock,

this presents us with a dilemma:

i) inference must respect the metaphysical necessities, and therefore the

above argument is valid; or

ii) we widen our notion of necessity and possibility by invoking a sense of

logical possibility according to which the inference is valid.

Pincock argues that we should adopt ii). His argument can be summarised

as the claim that if we adopt i), then all mathematical claims entail all other

mathematical claims due to the metaphysical necessary relationships between

them, which stops us from explaining how learning new mathematics can lead

to new inferences from scientific representations (Pincock 2012, pg. 34). The

idea is that, if the metaphysically necessary connections are respected, then

all mathematical claims are tautologous, and so all inferences including math-

ematics will be trivially valid. Thus when we perform an inference from a

mathematical scientific representation “all we wind up doing . . . is making ex-

plicit what was already implicit in the content of the original representation”.

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To see what sort of consequences this has for surrogative inferences, lets

look at the heat equation, (4.1), and how we use it to represent heat diffusion.

We specify all of the parameters of the equation’s schematic content to get

genuine content. This allows us to make predictions and test the accuracy of

the representation, which involves finding out what the accuracy conditions

actually are and looking to see if they occur in the target system. That is, we

solve the partial differential equations where variables have been given values

specific to the target system we are representing. Yet the solution to a partial

differential equation (if the equation is well formed) is unique and it is (sup-

posed) to take on unique values by metaphysical necessity. Pincock poses the

understanding of this inference to be problematic: are we supposed to under-

stand the metaphysical necessary connection between the partial differential

equation and the values of its solution to be a metaphysically necessary state-

ment about the iron bar and the way heat diffuses through it, for example?

Pincock’s solution to this problem, and motivation for adopting ii), is the

claim that “we can understand the content of the representation and yet not

be able to work out what the solution is. This suggests that the solution is

not part of the content of the representation”. This is supposedly clear, as the

solution is derived from metaphysically necessary truths, yet the necessity of

these truths is not “relevant to the conception of inference [Pincock] is . . . ar-

ticulating”. What is relevant, is the “nonlogical character of the mathematical

terminology”, as it is the nonlogical character of the terminology that “blocks

the entailment relation between the systems of equations and its solution. To

get genuine entailment, we need to add additional premises to the argument

corresponding to the features of the mathematical entities that are sufficient

to pin down the interpretation of the mathematical terms”.

This is where Pincock’s notions of intrinsic and extrinsic mathematics and

core conceptions come into play with regards to inference. The idea is that the

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content of our representations includes just the core conceptions of the intrinsic

mathematics, and it is these core conceptions which have a ‘fixed’ interpreta-

tion. As transitivity is part of the core conception of the ‘greater than’ relation,

the number of fish argument is valid without additional premises. However,

the solutions to partial differential equations are not part of the core concep-

tions of the partial differential equations themselves. Thus we need to employ

extra premises, extrinsic mathematics, in order to construct a valid inference

that the solution to the partial differential equations makes a prediction about

the target system. As these inferences require the extrinsic mathematics, the

triviality of the conclusion is removed, helping to alleviate the worry that our

mathematical scientific representations make claims that the world is necessary

in some way.

So why is this a problem for fitting the PMA into Contessa’s approach? The

way the approach was set out did not include any considerations of how math-

ematics can lead to valid inferences. The sense of validity Contessa seemed

to be concerned with was that of modus ponens : if A holds in the vehicle,

then B holds in the target; A holds in the vehicle, ∴ B holds in the target:

A → B; A; ∴ B. Given Pincock’s statements above, this appears to be too

simple an approach to inference. This difference in apparent simplicity can

be accounted for by looking at how the IC and the PMA deal with the heat

equation in terms of extra- and intra-vehicular inferences.

We can understand the heat example in a modus ponens argument form as

follows: we start with the claim that the solution of a PDE, S, represents the

heat diffusion at a particular point in the iron bar as having the magnitude S.

We solve the PDE to get a particular value for S (intra-vehicular inferences).

We then conclude that the heat diffusion in the bar at a particular point will

have the magnitude S (an extra-vehicular inference). This implies that the

notion of inference Contessa’s epistemic representation work with is only con-

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cerned with extra-vehicular inferences, such as predictions about magnitudes

and properties of the target system. I take this to be the same for the IC. This

understanding of inference is in contrast to Pincock’s, which concerns not just

predictions, but also how the predictions are reached, and so might threaten

the intra-/extra-vehicular inference distinction drawn above.

On the Contessa/IC view, solving PDEs are inferences that are internal to

the representing vehicle. One can solve PDEs with the content of representa-

tions on these accounts. But according to Pincock, solutions to PDEs are not

part of the content and so cannot be considered to be an intra-inferences. We

cannot consider them to be extra-inferences either as these are supposed to be

inferences from the vehicle to the target. Solving a PDE is an inference from a

piece of interpreted mathematics to another piece of interpreted mathematics,

which is then inferred from again to give the physical magnitude.

One possible solution is to consider the derivation of the solution to be

part of an extended extra-inference. One should consider the derivation of the

solution, and the extra premises needed (according to Pincock) in order to

turn this solution into a prediction, to be the extra-vehicular inference. This

fits both Pincock’s notion of inference and with the idea that what matters

for epistemic representation is an inference from a representational vehicle to

a target. There is nothing about this notion of extra-vehicular inference which

restricts it to only arguments of the modus tollens form, or at least not of

the A → B; A; ∴ B form where A is atomic. e.g. A could contain further

conditional statements (e.g. ((A ∧ B) → C) → D; ((A ∧ B) → C); ∴D). I

will not explore this solution further here. I take this solution to be good

enough to secure an understanding of Pincock’s account in terms of epistemic

representation.

I have shown above that Pincock’s view of inference originates from his

approach to representation. He starts from accuracy conditions, from which

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the inferential claims are supposed to follow, rather than concentrating on the

inferential claims and then attempting to explain how these claims result in

the evaluation of representations in terms of accuracy. This lead Pincock to

find the necessity of mathematical claims to be problematic, and finally to his

notions of intrinsic and extrinsic mathematics, and core conceptions.

The inferential approaches do not take the necessity of mathematical claims

to be problematic, or at least there is nothing in the literature to show that

they do. I think the reason for this is brought out in a footnote of Pincock’s,

where he rejects an idea put forward by a referee of his book, that “learning

mathematical truths may serve only to make us aware of valid inferences that

were already available from a given representation” (Pincock 2012, pg. 34, foot-

note 8). Roughly, I think the idea is this: mathematics allows us to perform

surrogative inferences, where the mathematical inferences we make are anal-

ogous to inferences we could make in the target; the mathematical inferences

‘mirror’ the inferences we could make with the target. This is obviously an

oversimplification, though this idea has some intuitive appeal, especially when

the inferential approaches are explicated in terms of morphisms between struc-

tures and given what the proponents of the IC state.27 A plausible response

to the necessity of mathematics on these inferential lead accounts is that while

the intra-vehicular mathematical inferences are necessary, the extra-vehicular

inferences are not, i.e. the interpretation of the conclusions of surrogative

inferences are not necessary.

27For example, that “by embedding certain features of the empirical world into a mathe-matical structure, it is possible to obtain inferences that would otherwise be extraordinarilyhard (if not impossible) to obtain” (Bueno & Colyvan 2011, pg. 352).

130

4.4 The Structural Mapping Accounts &

Faithful Epistemic Representation

Above I have argued that both the IC and the PMA can be accommodated

within Contessa’s approach, provided that we make some adjustments to the

first understanding of Contessa I put forward in §3.4. The PMA fits nicely into

the framework of Contessa’s analytic interpretation. The IC does not have the

separate relations required to supply epistemic representation, but does have

the resources to provide an account of faithful epistemic representation. The

adjustment required to accept accounts that provide faithful but not epistemic

representation was rejecting that the interpretation and gradable faithfulness

relations had to be distinct and accepting that one relation could play both

roles. The case of the IC, this is the interpretation mapping.

In this section I will provide broad answers to the questions concerning

faithful epistemic representation:

2. How does the account establish faithful epistemic representation? in-

cluding:

(2.a) How does it supply a gradable notion of faithfulness?

(2.b) How does it account for the relationship between faithfulness and

usefulness?

I have already given a broad answer to question (2): these accounts establish

faithful epistemic representation through structural relations between the ve-

hicle and the target. The details of what those structural relations are have

already been given in the introduction to each account, but below I will briefly

explain how these structural relations can answer question (2.a).

One of the arguments Suárez brought against the straw man isomorphism

account in his (2003) was the argument from misrepresentation. I argued that

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there are four types of misrepresentation: two types of mistargeting, mistaken

targets and non-existent targets; and two types of inaccuracy, empirically in-

adequate results, and due to abstraction and idealisation. Of these four types,

I held only non-existent targets and inaccuracy due to abstraction and ideal-

isation to be of particular relevance to accounts of scientific representation. I

have already sketched a solution to the problem of non-existent targets. The

problems caused by abstraction and idealisation require a lot of work to be

dealt with adequately. They have a direct bearing on the issue of faithfulness,

and on the relationship between faithfulness and usefulness. Contessa intro-

duces the idea that faithfulness and usefulness do not always coincide and can

in fact be opposed to each other due to considering Galilean idealisation (Con-

tessa 2007, pg. 130-131). An adequate answer to (2.b) will therefore require

a discussion of how each account deals with various types of idealisation. I

pursue this line of investigation in Chapter 6 for the IC, and in Chapter 7

for the PMA. I provide an answer to (2.b) for the IC in Chapter 6. However

an answer for the PMA has to wait until Chapter 8, as my investigation in

Chapter 7 finds answering the question difficult for the PMA (despite its initial

conformity with Contessa’s analytic interpretation).

4.4.1 The Inferential Conception

The structural relations employed by the IC are those of the partial structures

framework. These relations are often characterised as an ordered triple, R =

⟨R1,R2,R3⟩, where R1 is the set of n-tuples that hold for R, R2 is the set of

n-tuples that do not hold for R and R3 is the set of n-tuples for which is it not

known, or for which it is not defined, whether or not they belong to R.28

The first and strongest reason for adopting the partial structures framework

when attempting to account for the gradability of faithfulness is that the partial

28See §6.2.1 for a more detailed explanation of the partial structures framework.

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structures framework was introduced along with the notion of partial truth in

order to account for the partial information we have in science. By adopting

the view that scientific representations are (mostly) partially faithful epistemic

representations, in that they lead to some sound and some unsound inferences

to be drawn about their targets, one can see a direct parallel to how da Costa

& French (2003) claim scientific models can be partially true.

The faithfulness of the idealised vehicle can be measured by comparing

the Ri of the vehicle to the Ri of our target, in the same way the ‘degree’ of

similarity or approximation of models can be measured (da Costa & French

2003, pg. 50-51), (da Costa & French 1990, pg. 261). For example, the greater

the number of elements related by the R3 component, the more ‘partial’ the

structure is, and the less information we have, noting again that the R3 com-

ponent can be understood as containing those elements for which we do not

know whether statements are satisfied or not.

There is a problem in understanding the partial structures’ approach to

gradability in terms of only the ‘size’ of the R3 component. Once a partial

structure has been ‘filled out’, or ‘completed’, by mapping all elements of R3

to R1 or R2, we obtain the usual notion of structure. Yet this ‘full’ structure

might still produce unsound inferences; that is, the ‘full’ structure might still be

partially faithful. We have two options in response to this: either we are always

using partial structures, as a full structure would be the complete structure

of the situation under investigation; or we only obtain ‘full’ structures for

restricted domains. I think that the first of these responses is the more likely,

but do not have the space to argue for it here.29

The second benefit of adopting the partial structures framework is its (sup-

posed) ability to deal with the abstractions and idealisations present in the ma-

29A sketch of why I favour the first response is that scientific (mathematical) representa-tion seems to be one of the kinds of representation for which idealisation and abstraction,i.e. misrepresentation, is essential, and subsequently that restricted domains will involveidealisations and abstractions, i.e. that the second response will entail the first.

133

jority of scientific representations. A third, and related, benefit is the partial

structures framework’s capacity for dealing with inconsistencies in representa-

tional vehicles. Assuming that the world is not inconsistent (for the sake of

argument), an inconsistency in a representational vehicle would be a serious

case of unfaithfulness, a case that one might initially think results in a com-

pletely unfaithful representation. There are several instances of this not being

the case,30 most notably Bohr’s model of the atom.

As I said above, an investigation into how idealisations are accommodated

on the IC will wait until Chapter 6.31 I will also briefly point to how the

IC might deal with inconsistent representations, though I will also argue that

cases such as Bohr’s model of the atom should be dealt with by another part

of the Semantic View.

4.4.2 The Mapping Account

On Contessa’s approach, the partial faithfulness of a representation can be

measured by the number of sound compared to unsound surrogative inferences

a vehicle allows us to draw about a target. Unfortunately Pincock’s approach

to constructing his account of representation was to focus on accuracy condi-

tions rather than surrogative inferences, making the evaluation of his account

as an account of faithful epistemic representation difficult. Above I argued that

despite this difference in approach to constructing his account and the prob-

lems it raised for understanding inference on the PMA, the account could be

understood as an account of epistemic representation.32 As the PMA includes

a structural relation, which may take the form of any structural mapping or

30Or many cases if one adopts the position that idealisations can be understood as re-sulting in contradictions and therefore positing inconsistencies in the world Maddy (1992),Colyvan (2008).

31I will not address abstraction directly, as whatever is said for idealisation can be adaptedto accommodate abstraction.

32In fact, the PMA actually appeared to fulfil the criteria of Contessa’s analytic inter-pretation.

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mappings that include mathematical terms, the account should prima facíe be

capable of handling the gradability of faithfulness.

However, the focus on accuracy conditions appears to prevent this. This

is for two reasons. First, although Pincock talks about representations setting

accuracy conditions, he also talks about representations only being correct

if there is a structural relation M between the physical structure S and the

mathematical structure S∗, and incorrect if either of these structures do not

exist, or the relation does not hold (Pincock 2012, pg. 28). Second, Pincock

does not appear to follow a typical understanding of the notion of ‘accuracy

conditions’ as gradable. He cashes out ‘accuracy conditions’ in terms of placing

restrictions on the way the world must be for one to consider the representation

to be accurate, i.e. a strong claim about the structure of the target systems.

Pincock takes the descriptions of representations as “accurate” and “correct”

to be synonymous. The most relevant consequence of this is that whether a

representation is accurate/correct is a binary state of affairs: either it is correct

or it isn’t.

Fortunately we can understand the PMA as providing a gradable notion

of faithfulness. This involves emphasising the various notions of content and

the structural relation. These bring out two ways in which we can motivate a

gradable notion of faithfulness on Pincock’s account: one in line with the usual

understanding of accuracy; another in line with the way in which Contessa

aimed to provide an account of faithful epistemic representation by making use

of different types of structural relations to accommodate the difference in the

structure of target and vehicle (Contessa 2011, pg. 129-130). These two notions

of faithfulness come from an ambiguity in how one can understand ‘faithful’. As

defined above, Contessa takes a partially faithful epistemic representation to be

one which allows us to draw at least one sound inference from our representing

vehicle to our target. What these inferences are is left unspecified. Thus

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we can understand the inference to be a prediction that a magnitude has a

particular value, or that a target system has a particular property, or expresses

a particular behaviour and so on. Only the first of these types of inferences

can be understood in terms of the typical understanding of ‘accuracy’. I will

call this sort of inference numeric faithfulness. The other sort of faithfulness,

the sort that concerns the structural similarity of vehicle and target, can be

classed as structural faithfulness. Numerical faithfulness is just a consequence

of misrepresentation due to empirically inadequate results and so does not need

to be accommodated specifically by the PMA.

The structural faithfulness can be varied through the use of different struc-

tural relations and will involve considerations of the types of content involved.

The idea here is that while the choice of a particular structural relation might

produce an unfaithful representation, the choice of a different structural re-

lation will produce a more faithful representation. For example, the choice

between an isomorphism and an isomorphism with an error term in the heat

equation representation (Pincock 2012, pg. 31). The intentions of the agent

will play a role here, in selecting which structural relation to adopt.

Different structural relations and notions of content are involved in Pin-

cock’s approach to different types of idealisation. As such, a detailed account

of how the PMA can answer question (2.b) will require an investigation into

what Pincock has to say on idealisation. This will be pursued in Chapter 7.

4.5 Conclusion

In this chapter I have outlined the IC and the PMA, and argued that they

fit into Contessa’s approach of understanding scientific representation as a

subspecies of (partially faithful) epistemic representation. The IC cannot sup-

ply an account of epistemic representation, but this is not a problem as we

can adopt some other account of epistemic representation if required. As it

136

seems plausible that all scientific representations are partially faithful epis-

temic representations (in that they at one time facilitated the drawing of at

least one sound surrogative inference), we only need an account of faithful

epistemic representation to account of mathematical scientific representation.

I also argued that the IC had the resources to answer question (2.b). The

PMA satisfies the conditions for analytic interpretations. There were some

interpretative challenges to understanding the PMA as an account of faithful

epistemic representation due to the approach to inferences Pincock advocated

and his understanding of accuracy. These challenges could be met, though

how the PMA will answer question (2.b) still needs to be explained.

I will now turn to setting out the salient points of my case study, the rainbow.

137

5 | The Rainbow: An Introduction

5.1 Introduction

In this chapter I will outline the history of the scientific investigation into the

rainbow. The rainbow is an excellent case study for exploring issues related

to representation, in particular the role of idealisation. It provides several

different sorts of idealisations, as well as prompting interesting questions con-

cerning the interpretations of the mathematics involved. It has been used by

other philosophers to discuss issues such as reduction1 and mathematical ex-

planation.2 It has also been used by Pincock3 to discuss all of these issues and

so grants a great insight into his account of representation.

I will focus on two parts of the mathematics involved. The idealisation of

taking the ‘size parameter’4 to infinity, and the introduction of abstract math-

ematics in the Complex Angular Momentum (CAM) approach to finding an

approximate solution to the Mie solution. The β →∞ idealisation is involved

in a ray-theoretic explanation of the colour distribution of the rainbow, while

the CAM approach is involved in an explanation of the supernumerary bows,

an interference pattern that is sometimes seen inside the primary rainbow.

These two pieces of mathematics provide useful examples for gaining further

1Batterman (2001)2Batterman (2010) and Bueno & French (2012).3Pincock (2012) Chapter 11.4The size parameter is a dimensionless variable β that describes the relationship between

the wave number of the light incident on the water droplet and the radius of the waterdroplet.

138

insights into the Inferential Conception and Pincock’s Mapping Account. I

will return to these examples in subsequent chapters, but outline them here so

that the reader gains a familiarity with them before I put them to philosophical

work.

The history of the investigation into the rainbow can be roughly split into

three. First, there was the ray theoretic approach that was able to provide

apparent explanations for the angle of the rainbow and the distribution of

colours within it. Here, light is represented as geometric rays and the rainbow

is explained to be due to internal reflections of the these light rays within

the water droplets. Next came the wave theoretic Mie solution. Mie was

able to derive an exact solution of the scattering of an electromagnetic light

wave by water droplets using Maxwell’s equations. Unfortunately the Mie

solution, expressed as a sum of partial wave terms, converged too slowly to

provide practical results. This prompted a search for a way to transform

the solution so that an approximate, but faster converging, form of the Mie

solution could be found. As part of this search, an analogy was made with the

scattering of quantum particles: light was represented via the mathematics

used to represent the complex angular momentum of quantum particles. This

move to the complex plane allowed for an approximate solution to be found,

which now gives an accurate result for a very large range of physical situations.

I will discuss the first and third of these parts of the investigation in much more

detail below. I will only provide a sketch of the Mie solution. Full mathematical

details can be found in Appendix A.

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5.2 The Ray Theoretic Approach to The

Rainbow Angle and Colours

The two most striking features of a rainbow, that it always appears at an angle

of 42° to the observer and with a colour distribution of violet, blue through to

orange and red from bottom to top, can be accounted for by the ray theory of

light. Here light is considered to be a ray, rather than an electromagnetic wave.

As such, it lacks various wave features: there is no wavelength, frequency and

so on.

changes as we consider rays which strike the drop. As Fig. 2 shows, apath that involves an initial refraction, followed by a reflection onthe back of the drop, followed by a second refraction can lead to aray that will be deflected backwards and downwards towards theobserver. The angles in question here are the initial angle ofincidence yi and the initial angle of refraction yr . Snel’s law andthe geometry of the circle allow us to determine the total angle ofdeflection D using just yi:

D¼ 1803þ2yi#4sin#1 1n

sinyi

! "ð1Þ

The basic idea of our explanation of (a) is that (1) tells us that muchof the sunlight that hits the raindrop at angles 03oyio903 willhave D at or close to D¼ 1383. This corresponds to the observedangle of a¼ 423 ¼ 1803#D that we are trying to account for. Thisproposal can be made more precise by noting that D obtains aminimum, and so a obtains a maximum, as we move from yi ¼ 03 to903 when D¼ 1383. Furthermore, this minimum corresponds to alarge amount of light because many yi lead to this value for D orsomething close to it. The 423 is the maximum value for a, soaccording to this explanation no sunlight from these rays appearsabove the primary rainbow.

So far no mention has been made of the colors of the rainbow.That means that our explanation of (a) is clearly not an explanationof (b). But a small addition to our account of where the rainbowappears is sufficient to rectify this gap. In the above argument wetreated only the case where n¼ 4

3. Treating n as a constant issufficient to explain where the rainbow appears, but a moreaccurate perspective on n is that it is a variable which dependsnot only on the media involved, but also on the color of lightundergoing the refraction. At one end of the visible spectrum, thevalue of n for red light is roughly 1.3318, while for violet light at theother end of the spectrum, n is approximately 1.3435. This slightchange in n can be used to account for the angular spread betweenthe colors of the rainbow and their order. For example, our newvalue for n for red light yields a value of D to be 137:753, while withthe new value of n for violet D becomes 139:423. As a result, a isgreater for red than for violet and the red band appears above theviolet band in the sky (Adam, 2003, p. 89).

While this is clearly an explanation for (b) which can be easilycombined with our explanation of (a), it is not clear that this is thebest explanation of (b). Furthermore, when we press on thisexplanation for (b) and see an explanation which has a betterclaim on being the best explanation for (b), we will encounter theworry that our explanation of (a) is problematic as well. To seethe limitations of our explanation of (b) notice that we have treatedthe link between color and n as a series of brute facts. It is here thatan appeal to the wave theory of light promises to do better. For oncewe make the link between the decreasing wavelength of light andthe change in color from red through the visible spectrum to violet,we can account for the change in n. According to the wave theory oflight what is going on here is the more general phenomena of lightdispersion where the behavior of light can be affected by thewavelength of the light wave. A beam of white light is thenrepresented as a superposition of waves of various wavelengthswhich are separated as each component wave is refracted at aslightly different angle. Going further, we can use this waverepresentation to account for the physical claims which the rayrepresentation takes for granted. These claims include Snel’s Lawand the law of reflection.

On this view the best explanation of (b) takes light to be made upof electromagnetic waves. This is the best way to make sense ofthe colors of light and how the colors come to arrange themselves inthe characteristic pattern displayed by the rainbow. Now, though,we have a reason to doubt the explanation we have given for(a) because this explanation presented light as traveling alonggeometric rays. Without some relationship to our wave represen-tation of light, this account of (a) seems to float free of anything weshould take seriously. This point raises an instance of the centralquestion of this paper: how can we combine the resources of theclearly incorrect ray representation with the correct wave repre-sentation to provide our best explanations of features of therainbow? The answer that I would like to suggest follows imme-diately from a certain approach to idealization. This is that the rayrepresentation we have deployed need not require the assumptionthat light is a ray. It may assume only that in certain circumstancessome features of light can be accurately represented using the rayrepresentation. But what are those circumstances and how do theyrelate to these features? A first attempt to answer this questionmight draw attention to one of the mathematical relationshipsbetween the two representations. This is that the ray representa-tion results from the wave representation when the wavelength istaken to 0. Another way to put this makes use of the wave number k,which is equal to 2p divided by the wavelength. We then can putthis limit in terms of the wave number going to infinity. This leadsto a representation in which the wave-like behavior of light iscompletely suppressed. We can then track the path of the lightusing geometric lines and deploy simple rules like the law of

Fig. 2. Nahin (2004, p. 181).

Fig. 1. A rainbow.

C. Pincock / Studies in History and Philosophy of Modern Physics 42 (2011) 13–22 15

Figure 5.1: The basic geometry of a ray incident on a spherical water drop. Thesingle internal reflection leads to the primary rainbow. O is the centre of the waterdrop, i the angle of incidence, r the angle of reflection. Reproduced from Nahin(2004).

The angle of the rainbow can be explained through applying the claim

that the angle of incidence equals the angle of reflection and Snell’s Law to

the path of a light beam that hits a water droplet. The light beam undergoes

refraction, then internal reflection on the back surface of the water droplet,

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and finally refraction when exiting the raindrop. Snell’s law and the geometry

of the circular water droplet allows us to claim that the total angle of deflection

of the light ray, D is equal to:

D = 180° + 2θi − 4 sin1 1

nsin θi (5.1)

where θi is the angle of incidence of the light ray on the water droplet. D, θi,

and α, the observation angle, are those in Figure 5.1. The rainbow occurs at

the angle that the majority of the light is refracted out of the water droplet

(see Figure 5.2). That is, the rainbow occurs at the angle where D is minimum

and α, as given in (5.2), obtains a maximum, for the light hitting the water

droplet at angles 0° < θi < 90°.

α = 180° −D

= 2θi − 4 sin1 1

nsin θi (5.2)

We find that the maximum value α may take is 42°, hence the angle of the

rainbow. Further, as this is the maximum angle, no light will be seen at a

greater angle than this: no light is refracted and reflected above the primary

bow.

In order to account for the distribution of colours, we note that n, the

refractive index, can vary depending upon the colour of the light incident on

the water droplet. The above calculation for the maximum value of α, at 42°,

assumed that the light was white, giving a refractive index of n = 43 . But

white light is composed of many colours of light, each of which has a different

refractive index. Thus we can run the calculation for the maximum angle of

reflection using the refractive index of red, orange, through to violet light and

establish the maximum value α takes for each colour. Here we find that α for

red light is 42.25° and for violet light is 40.58°. As θα red is greater than θα violet,

141

the red band of light will be above the violet.5J.A. Adam /Physics Reports 356 (2002) 229–365 235

Fig. 2. The paths of several rays through a spherical raindrop illustrating the di!erences in total de"ection anglefor di!erent angles of incidence (or equivalently, di!erent impact parameters). The ray #7 is called the rainbow rayand this ray de#nes the minimum angle of de"ection in the primary rainbow. (Redrawn from [4].)

deviation angle increases again. The ray of minimum deviation (#7 in Fig. 2) is called therainbow ray. The signi#cant feature of this geometrical system is that the rays leaving the dropare not uniformly spaced: those “near” the minimum deviation angle are concentrated aroundit, whereas those deviated by larger angles are spaced more widely (see also Fig. 3).Put di!erently, in a small (say half a degree) angle on either side of the rainbow angle (≃

138) there are more rays emerging than in any other one degree interval. It is this concentrationof rays that gives rise to the (primary) rainbow, at least as far as its light intensity is concerned.In this sense it is similar to a caustic formed on the surface of the tea in a cup when appropriatelyilluminated. The rainbow seen by any given observer consists of those deviated rays that ofcourse enter his eye. These are those that are deviated by about 138 from their original direction(for the primary rainbow). Thus the rainbow can be seen by looking in any direction that isabout 42 away from the line joining one’s eye to the shadow of one’s head (the antisolarpoint); the 42 angle is supplementary to the rainbow angle. This criterion de#nes a circular arc(or a full circle if the observer is above the raincloud) around the antisolar point and hence allraindrops at that angle will contribute to one’s primary rainbow. Of course, on level ground, atmost a semi-circular arc will be seen (i.e. if the sun is close to setting or has just risen), andusually it will be less than that: full circular rainbows can be seen from time to time at highaltitudes on land or from aircraft. In summary, the primary rainbow is formed by the de"ectedrays from all the raindrops that lie on the surface of a cone with vertex (or apex) at the eye,axis along the antisolar direction and semi-vertex angle of 42. The same statement holds for thesecondary rainbow if the semi-vertex angle is about 51 (the supplement of a 129 deviation).

Figure 5.2: Diagram demonstrating that light groups around the minimum angle ofdeflection. The ray numbered 7 is the ‘rainbow ray’ and defines the minimum angleof deflection, i.e. the rainbow angle θR. Reproduced from Adam (2002).

While the angle of the rainbow and the colour distribution can be explained

by a purely ray theoretic account, as above, this explanation can be seen as

lacking in certain respects. For example, the link between the colour of light

and refractive index is one of stipulation, or being treated as a “brute fact”

(Pincock 2012, pg. 226). Pincock argues that not only is the explanation for

colour problematic in this way, so too is the explanation for the angle: the pure

ray theoretic explanation leaves several parts of the explanation underdevel-

oped and unexplained itself. The solution is, roughly, that we can do better by

adopting a wave theoretic position and obtaining the ray theoretic explanation

through a transformation of the wave theoretic mathematics. More precisely,

we should adopt a particular idealisation of the wave theory that results in a

ray conception of light. This would then allow us to use the wave theoretic ex-

planation for why refractive indices vary by the colour of light: refractive index

5The smaller angle for the violet light puts it on the inside of the reflected light cone,the red on the outside. Hence, for the top half of the cone from our line of sight, red is atthe top of the rainbow, while violet is at the bottom.

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depends upon the wavelength of the incident light, and wavelength determines

the colour of light.

One way of performing this idealisation would be to take the wavelength

of light to 0, which can also be understood as taking the wave number, k = 2πλ ,

to infinity. Unfortunately this idealisation leads to a representation that loses

the most important piece of the wave theory for accounting for the varied

refractive indices, the wavelengths of the differently coloured light. However,

there are still some wave like behaviours at this limit. Batterman (making use

of Berry (1981), and so providing a derivation of the ray representation that is

amenable to catastrophe optics) claims that a ray representation obtained by

defining rays to be normals to geometrical wavefronts displays such a feature.

Here, a ray is defined as the gradient of the optical distance function, which

is the Euclidean distance between the point of wavefront we take the ray to

be normal to, and some point P , where we are attempting to approximate the

wave via rays.6 There will be many rays passing through this point due to the

curvature of the wavefront, so it would be natural to consider the amplitude

and phase of the resultant wave to be due to contributions from all of these

waves. Using Fermat’s principle, defining the phase of the nth ray to be 2π

times the optical distance, φ, from the wavefront to P in units of wavelength,

and approximating the wave intensity by considering the flux through an area,

we obtain the shortwave approximation:

ψ(R, z) ≈∑n

∣det∂rn

∂R(R, z)∣

12

eikφn(R,z) (5.3)

Berry7 states that (Berry 1981, pg. 519-520):

[a]s a shortwave (or, in quantum mechanics, semiclassical) approxima-

tion, [(5.3)] has considerable merit. Firstly, it describes the interference

6See Batterman (2001), Figure 6.4, pg. 85.7As quoted by Batterman (2001), pg. 88.

143

between the contributions of the rays through R, z. Secondly, it shows

that wave functions ψ are non-analytic in k as k → ∞, so that short-

wave approximations cannot be expressed as a series of powers of 1k , i.e.

deviations from the shortwave limit cannot be obtained by perturbation

theory. Thirdly, in the shortwave limit itself [k = ∞], ψ oscillates in-

finitely fast as R varies and so can be said to average to zero if there

is at least some imprecision in the measurement of R in the intensity

∣ψ∣2, the terms from [(5.3)] with different n average to zero in this way,

leaving the sum of squares of individual ray amplitudes, i.e.

∣(R, z)∣2 =∑n ∣det∂rn

∂R(R, z)∣ (5.4)

when k = 8 and this of course does not involve k

This case suggests that it is possible to find a ray representation that still has

wave features present at the limit. However, this particular way of obtaining

the ray representation fails, not only because we’ve taken k → ∞, but also

because the amplitude it describes becomes infinite on a caustic. Rainbows

can be considered as caustics, so the shortwave approximation fails exactly

where it is needed.8

We therefore need to find an idealisation that gives us a ray conception

of light, yet allows us to refer to the wavelength of light in some way. The

suggestion is that we use the size parameter, β. This is a dimensionless pa-

rameter, defined as the product of the radius of the water drop, a, with the

wave number: β = ka = 2πλ a. Provided that the water droplet is large enough in

comparison to the wavelength of the incident light, we can use this idealisation

to describe the direction the light travels in with a ray. The idea is that the

distance between the wave crests is sufficiently small compared to the radius of

8See Batterman (2001, §6.3 for this discussion. The shortwave approximation fails interms of being unable to tell us how the intensity changes or the wave pattern (the fringespacings) on or near a caustic as k →∞ (Batterman 2001, pg. 88).

144

the water droplet, such that they appear as if they form a continuous straight

line. Thus we can still appeal to the wavelength of the light to account for the

differences in refractive index.

Pincock claims that “when [Batterman] is being careful”, he refers to this

limit (as opposed to the k →∞ limit). Further, we can see that van de Hulst

(1981) makes use of this limit. van de Hulst pioneered the use of the impact

parameter for light scattering, which associates a partial wave with an incom-

ing ray. This allows one to say that one has localised a partial wave to a ray.

Thus I propose that any ray representation thats use of the notions of impact

parameters or localised waves will make use of the β →∞ idealisation to obtain

its ray representation. Finally, in comparing the accuracy of a ray represen-

tation to the Mie solution, Grandy Jr makes use of the β → ∞ idealisation

from the Mie solution (Grandy Jr 2005, §4.3). As the Mie solution for the

rainbow is a wave theoretic solution, it is clear to me that we should adopt an

idealisation that obtains the ray representation from the Mie solution.

How one justifies this idealisation, and how one accounts for the structural

moves will depend upon which account of representation one adopts. In the

next two chapters, this idealisation will be appealed to. In the case of the IC,

it will be used as an example of the introduction of surplus structure due to

a limit. For the PMA, it will be used to discuss Pincock’s position of meta-

physical agnosticism. This is a position Pincock offers as a third response

to the predictive success of singular perturbations, after rejecting an instru-

mentalist position towards the mathematics and the metaphysical positions of

reductionism and emergentism.

5.3 The Mie Solution

The Mie solution is arrived at by solving the Maxwell equations for the scat-

tering of a plane wave of monochromatic light by a homogeneous sphere (the

145

water droplet). First, the Maxwell equations are solved for waves in an arbi-

trary medium. These general results are then applied to the specific case of

scattering of a plane wave by a homogeneous sphere. We find the potentials

inside the sphere, and for the incident and scattered waves outside the sphere,

and match coefficients. The scattered wave is then solved in the far field zone

to give the Mie solution, (5.5).9

Sj (β, θ) =1

2

∞∑n=1

[1 − S(j)n (β)] tn (cos θ) + [1 − S(i)

n (β)]pn (cos θ) (5.5)

(i, j = 1,2i ≠ j)

The Mie solution gives the scattering amplitudes of the scattered wave in

terms of the angular functions pn and tn and S-matrix elements Sjn, where the

S-matrix give the scattering of the partial waves in terms of phase shifts.

5.4 Supernumerary Bows & The CAM

Approach

The explanation of the angle and colour distribution that depended on a ray

conception of light, even the conception arrived at through the β → ∞ ideal-

isation, is incapable of providing an explanation of the supernumerary bows.

These are bands of constructive and destructive interference sometimes seen

inside the primary rainbow. The Mie solution is capable of providing an ac-

count of how these come to be, but the partial wave sum converges very slowly.

It needs terms in the order of the size parameter, typically over 5000 terms, to

produce a result which is sufficiently accurate, that allows for the later terms

to be disregarded.

The solution to this problem is to transform the Mie solution in a form9Full details of this derivation can be found in the Appendix, A, which draws on Nussen-

zveig (1992), Liou (2002), Adam (2002) and Grandy Jr (2005).

146

that converges much quicker, and thereby allowing us to ‘extract the physics’

from the solution. The approach taken is to apply the Poisson sum formula to

individual terms of the Debye expansion. We apply it to each term individually

as applying it directly to the Mie solution as a whole results in a series that

converges at about the same rate as the Mie solution itself (as will be shown

below). The Poisson sum formula, (5.6), allows for a infinite sum over a real

function to be transformed into an integral over complex variables.

∞∑l=0

ψ (Λ, r) =∞∑

m=−∞(−1)m

∞

∫0

ψ (Λ, r) e2imπΛ dΛ (5.6)

This move facilitates the analogy to quantum scattering and the description

of light as possessing a complex angular momentum. While a straightforward

application of the Poisson sum formula does not require the move into the

complex plane, this is required for generating the asymptotic approximations

that are essential for the CAM approach. In summarising the history of this

approach, Nussenzveig explains that Watson proposed a transform based on

equation (5.7) (which can be shown to be equivalent to the Poisson sum for-

mula).

∞∑l=0

φ(l + 1

2, x) = 1

2 ∫C φ (λ,x) e−iπλ

cosπλdλ (5.7)

This includes a contour integral, the path for which was chosen to be about the

real axis.10 Once in the complex plane, one can deform the paths over which

one is performing integrals, provided that one takes into account any possible

singularities. The singularities met are the poles of the function. Poles are

features of complex functions. The necessary concepts required to understand

what they are outlined in §A.6. Briefly, poles are found when the bn coeffi-

cients of the Laurent series expansion of f(z) are non-zero. The coefficient

10See Nussenzveig (1992), Figure 6.2, pg. 49.

147

b1 is described as the residue of f(z). When subjecting the scattering func-

tions to the CAM approach, they are transformed into a form that include a

residue series ; they include a sum over all poles in the complex plane and the

residue rn at pole λn. See, for example, Grandy’s discussion of the application

of the Watson transform to the partial wave series for quantum mechanical

scattering:11

[O]ne rewrites the partial-wave series as a contour integral:

f (E, q2) = 1

2i∫C

(2l + 1) fl (E)Pl (− cos θ)sinπl

dl (5.8)

where the contour C [surrounds the real axis, crossing the origin]. An-

alytic continuation is effected by deforming the contour to the vertical

and closing it to the left in two quarter circles to avoid the bound-state

poles. In doing so the contour ‘sweeps across’ a number of Regge poles,

thereby picking up the corresponding residues. . . . The result of this

continuation is the Watson transformation of the scattering amplitude:

f (E, q2) = 1

2i∫C′

(2l + 1) fl (E) Pl (− cos θ)sinπl

dl + πn

∑i=1

βi (E)Pηi(E) (− cos θ)

(5.9)

where the βi (E) are the residues of the integrand at the Regge poles with

the Legendre function separated out. The first term on the right-hand

side of (5.9) is referred to as the background integral, and the second is

the residue series.

The poles of the scattering function in the complex plane are known as

Regge poles. We concentrate on the Regge poles as the motivation behind the

CAM approach, specifically the path deformations they allow us to perform.

The idea is to (Nussenzveig 1992, pg. 62):

11Grandy Jr (2005), pg. 47-49.

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concentrate dominant contributions to the integrals into the neighbor-

hood of a small number of points in the λ plane, that will be referred

to as critical points. The main types of critical points are saddle points,

that can be real or complex, and complex poles such as the Regge poles

The saddle points (points where a curve in one direction has a maximum and

a curve in another has a minimum) are the major contributions to the back-

ground integrals (when the integrand has saddle points) (Nussenzveig 1992,

pg. 64). The contributions of these saddle points is found through the method

of ‘steepest descent’. This method will be outlined briefly below. It is found

to be wanting in the initial application of the CAM approach, which leads to

the development of the CFU method.90 The Debye expansion

Im.< . h ic t X X4 x

3 J 2 X I .x.x.x.x .. .x ... x ... JC ••• x .. .X····X···•·:'-x ... x ..

i-/1 0 fJ a x '.i.

St lt X ic : h'

Fig. 9.2. Regge poles for N > I and a given polarization. X poles; I narrow resonances; 2 broad resonances; 4 surface waves.

behavior is determined almost entirely by the geometry of the surface. The other class of poles, distributed above the real axis along a

curve j that extends to A. = a, is affected by the potential in the internal region. The poles located in region 1, between A. = f3 and A. = a, are associated with the narrow resonances formed in the domain between T and Bin fig. 9.1(a). They may approach very close to the real axis. These poles and their physical effects will be discussed in Chapter 14.

The poles in region 2, to the left of A. = /3, are farther from the real axis, corresponding to broader resonances above the top of the cusped barrier in fig. 9.1(a). To the left of the imaginary axis, in region 3, the poles tend to approach the negative integers.

The separation between the poles along the curve j is of order unity, so that the number of resonance-type poles in the first quadrant is 0 (/3). Since their imaginary parts are not large, a residue series over these poles would converge about as slowly as the partial-wave series. Thus, it would not be advantageous to apply the Poisson sum formula directly to the total scattering amplitude (5.6).

9.3 The Debye expansion

The convergence difficulties found, in contrast with the impenetrable sphere problem, when one tries to apply the CAM method directly to the total amplitude arise from the penetration of the waves inside the sphere. In the geometrical-optic limit, as illustrated in fig. 2.1, the interaction with the sphere is described as a sequence of interactions with its surface

Figure 5.3: Regge poles for N > 1. The x represent the poles. Those in region 1are those for narrow resonances, 2 for broad resonances, and 4 for surface waves.Reproduced from Nussenzveig (1992).

Applying the Poisson sum formula directly to the Mie solution and extend-

ing the result into the complex plane gives us the scattering function S(j) (λ,β),

given in (5.10). The Regge poles are the roots of 1β −mvjα = 0, the de-

nominator of S(j). These Regge poles can be placed into two classes. The first

of these are along the curves 4 and 5 in Figure 5.3. These are associated with

surface waves. The first class of poles are of relevance below, once we have

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applied the Debye expansion, while the second class of poles are of immedi-

ate relevance. These are those found above the real axis along the curve j in

Figure 5.3. The separation of these poles is of the order 1, and their imag-

inary part is not large. As there are as many poles as O(β) in the relevant

part of the complex plane for the residue series, the series over these poles

will converge about as slowly as the partial wave series in the Mie solution.

Clearly, we need a form of the Mie solution that produces a residues series that

converges quicker than the partial wave series in the Mie solution, so applying

the Poisson sum formula as we have, directly to the Mie solution, is not the

way to go. Thus we turn to the Debye expansion of the Mie solution.

S(j) (λ,β) = −ζ(2)λ (β)ζ(1)λ (β)

[2β −mvjα1β −mvjα

] j = 1,2 (5.10)

[jz] = ln′ ζjλ(z) +1

2z(5.11)

[z] = ln′ψλ +1

2z(5.12)

The Debye expansion involves representing the multipole of index n in

the partial wave sums (found in the scattering amplitude equations, (A.123) -

(A.125)) in terms of incoming and outgoing spherical waves that are partially

transmitted and partially reflected at the surface of the sphere (rather than

the standing waves inside the sphere, and the scattered plane waves outside

the sphere) (Grandy Jr 2005, pg. 144) (Adam 2002, pg. 283). As the Debye

expansion involves modelling the waves as partially transmitted and reflected

waves, we write the scattering function in terms of spherical reflection and

transmission coefficients (themselves expressed in terms of Henkel and Bessel

functions). For example, the external spherical reflection coefficient is given in

(5.14).12 The rest of the coefficients are given in equations (A.145) - (A.148)

in the Appendix.

12Note the difference in the definitions of [j] and [jz]. This difference is because theseequations have not yet been extended to the λ plane.

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S(j)n = R22

n + T 21n

∞∑p=1

(R11n )p−1

T 12n (5.13)

R22n (β) ≡ −ζ

(2)n (β)ζ(1)n (β)

[2β] −mvj [2mβ][1β] −mvj [2mβ]

(5.14)

S(j) (β, θ) = Sj,0 (β,0) +P

∑p=1

Sj,p (β, θ) + remainder (5.15)

[jz] = ln′ ζjλ(z) +1

2z−1 (5.16)

[z] = ln′ψλ +1

2z−1 (5.17)

The Debye expansion of the total scattering amplitude, in terms of the

spherical reflection and transmission coefficients in (5.13). It is also given in

terms of the scattering amplitudes, is given in (5.15). Sj,0 is the direct reflection

term, Sj1 is the direct transmission term, and Sjp, where p ≥ 2 corresponds to

transmission following p−1 internal reflections. We now apply the Poisson sum

formula to each individual term. It is the p = 2 term that we concentrate on,

as it is responsible for the primary rainbow (as it is the term for one internal

reflection). We find a rapidly converging residue series for these terms. Here,

rather than Regge poles, we talk of Regge-Debye poles (to reflect that we

are dealing with the Debye expansion of the Mie solution). These poles are

found from the roots of the equation 1β−mvj2α = 0, which is the common

denominator of all of the spherical transmission and reflection coefficients. This

equation differs from the one used to find the Regge poles when we applied the

Poisson sum formula to the Mie solution by the substitution α→ 2α, which

reflects the change from standing waves to ingoing waves within the sphere.

When we look at the poles for the Debye expansion, we find only poles of the

first type outlined above (Nussenzveig 1992, pg. 93-94). We also obtain a new

set of poles, found in the second quadrant of the complex plane, which have a

large imaginary part compared to the corresponding poles in the first quadrant.

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One can see the Regge-Debye poles in Figure 5.4. Thus we expect that the

residue series contributions from the Regge-Debye poles to become rapidly

damped as n increases, and so we can obtain rapidly convergent asymptotic

expansions for each term of the Debye series through the CAM approach.94 The Debye expansion

lm,\

* X x X ,j_U> X n x. x x x 'i

Re-\ -a -(J 0 fJ a

Fig. 9.4. Regge-Debye pole distribution for N > I and a given polarizationj.

Regge-Debye poles are rapidly damped as n increases. It follows that the CAM method leads to rapidly convergent asymptotic approximations for each term of the Debye series.

The absence of resonance effects in individual Debye terms and the presence only of surface-wave type pole contributions agree with the physical interpretation of the Debye expansion as a description based exclusively on surface interactions.

The other question that has to be discussed is how fast the Debye series itself converges. While a fuller discussion of this question must wait until we have developed further insight into the behavior of the various terms, we can anticipate some of the results.

According to (9.7), successive terms of the Debye expansion differ by an additional factor pV)(A,J) in their Poisson transform integrands. For real A and N, by (9.9),

(9.12)

so that the rate of convergence depends on the magnitude of the internal spherical reflection coefficient within the A range that yields significant contributions.

Contributions from real saddle points are associated with geometrical-optic rays, for which the Debye terms are in one-to-one correspondence with the scattered rays 0, 1, 2, ... shown in fig. 2.1. The angle of incidence 81 in this figure is related to the saddle point I by

(9.13)

where we have employed the localization principle. The rate of convergence of real saddle-point contributions is the same as that of the multiple internal reflection geometrical-optic series, determined by the Fresnel internal reflection coefficient at the interface.

Figure 5.4: Regge-Debye poles for N > 1. Reproduced from Nussenzveig (1992).

When the CAM approach was first developed, it was non-uniform for all

m > 1 and for the domain 0 ≤ θ ≤ π (Nussenzveig 1992, pg. 101). The domain

θR < θ < θL is covered twice (i.e. has incident light refracted at this angle from

two separate incident rays/waves), where θR is the rainbow angle and θL is

the maximum deflection angle (at θL = 4 cos−1 1m), leading to a subdivision at

the geometrical-optic level into three angular regions: the 0-ray (geometrical

shadow) region 0 ≤ θ < θR; the 2-ray region θR < θ < θL; and the 1-ray region

θL < θ ≤ π. In each region, the Poisson transformed scattering amplitude is

reduced to a background integral, which typically has contributions dominated

by the saddle points and a Regge-Debye pole residue series (Nussenzveig 1992,

pg. 102). We find that as θ increases from θ = θL towards θ = θR, there are

two saddle points which converge and meet at θ = θR. The rainbow is thus de-

scribed in the λ plane as the collision between two saddle points. A reason for

the non-uniformity of the CAM approach is that the method for establishing

the contributions due to the saddle points to the background integral breaks

down when the ranges of the saddle points overlap. The notion of the range of

a saddle point is introduced as, for asymptotic purposes, one does not always

have to pass through the exact saddle point, but through an approximation

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to it. The notion of the range of the saddle point is supposed to clarify what

deviations are allowable. de Bruijn offers the following informal definition: “if

ζ is a saddle point of the function ψ, then the range of ζ is a circular neigh-

bourhood of ζ, consisting of all z-values which are such that ∣ψ′′(ζ) (z − ζ)2∣ is

not very large” (de Bruijn 1958, pg. 91).

The Chester-Friedmann-Ursell (CPU) method is an answer to the question

“what happens in the saddle-point method when two saddle points approach

one another?” (Nussenzveig 1992, pg. 105). The original approach to provid-

ing an answer to this question involves the method of steepest descent. One

considers a general complex integral, such as in (5.18). Here the integral is

over some path in the w plane, where κ is an asymptotic expansion parameter

(like β) and is large and positive, and ε is an independent parameter (like θ).

If the integral is dominated by a single saddle point w = w(ε), around which

f and g are regular, one can approximate in this neighbourhood (5.19) where

f ′′w denotes the second derivative with respect to w.

F (κ, ε) = ∫ g(w)e[κf(w,ε)] dw (5.18)

f(w, ε) ≈ f(w, ε) + 1

2f ′′w(w, ε)(w − w)2 (5.19)

f(w, ε) = 1

3u3 − ζ(ε)u +A(ε) (5.20)

Choosing the steepest decent through w alters the relevant part of the integral

to a Gaussian type. An asymptotic expansion of the integral can be obtained

by adopting a change of variables that makes the exponent into an exact Gaus-

sian, followed by a power series expansion of g around the saddle point and

integration term by term (Nussenzveig 1992, pg. 106). If there are two saddle

points, w′ and w′′, as in the 2-ray region, then their contributions can be added

independently provided that their ranges do not overlap, which is what occurs

when the two saddle points collide near θR. Thus, we need to adopt the CFU

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method to establish what happens at θR.

The basic idea of the CFU method is to transform the exponent of (5.19)

into an exact cubic through a change of variables, e.g. (5.20). There the

two saddle points have been transformed into ±ζ 12 (ε). The crucial point is to

introduce “a mapping that preserves the saddle point structure”. Substituting

into (5.18) and suitable expansion of the integrand allows one to obtain an

equation for F (κ, ε) in terms of the Airy function, Ai.13 Making the relevant

substitutions to apply the general equation (5.18) to the rainbow case allows

us to write the dominant contribution to the third Debye term in the rainbow

region in the form of (6.1):

Sj2(β, θ) ≈ β76 e[2βA(ε)] [cj0(ε) + β−1cj1(ε) + . . . ]Ai [(2β)

23 ζ(ε)] (5.21)

+ [dj0(ε) + β−1dj1(ε) + . . . ]β−13Ai′ [(2β) 2

3 ζ(ε)]

where the CFU coefficients cjn(ε) and djn can be expressed in terms of the

Fresnel reflection and transmission coefficients and their derivatives of various

orders with respect to the angle of incidence, evaluated at the exactly known

saddle points θ′1, θ′′1 (the order of the derivatives increases with n) (Nussenzveig

1992, pg. 108). The CFU method allows for the CAM approach to be applied

uniformly for the whole range of θ.

5.5 Philosophical Responses

The two main contributions to the solutions found via the CAM approach are

those provided by the critical points, the saddle points and the Regge-Debye

poles. Throughout the derivations, we find references to the Regge-Debye poles

13The Airy function originates from Airy’s treatment of the rainbow, which made useof a cubic wave front. For more details see (Adam 2002, §2.1), (Nussenzveig 1992, §3.2),Grandy Jr (2005, Appendix B), etc.

154

as being interpreted in terms of surface wave contributions.14 This can either

be understood wave-theoretically, as waves that travel along the surface of the

drop rather than enter into the water droplet, or ray-theoretically as rays that

take shortcuts by being critically refracted along the edge of the water droplet

before being critically refracted back into the droplet. The saddle points are

often said to represent rays,15 though real saddle points can also be interpreted

in terms of points of stationary phase.16.

Exactly which interpretation one should adopt, and how that interpretation

is to be described in terms of which piece of mathematics is related to some-

thing else (more mathematics, the world, etc.) will vary according to which

account of representation one adopts. For the IC, the CAM approach provides

another, though different, approach to surplus structure than the β →∞ ideal-

isation appears to. The CAM approach is a clear case of moving to a different

mathematical model which will have to be related to the first mathematical

model before a physical interpretation can be found. The application of the

Poisson sum transformation and the work with saddle points and Regge-Debye

poles in the complex plane appears to be a use of surplus structure. The saddle

points and Regge-Debye poles will have to be mapped back to something in

the wave model before they can be given a physical interpretation. The CAM

approach is therefore an example of the iterative version of the IC, outlined in

Bueno & French (2012)

Pincock aims to distinguish his position from Batterman’s. Pincock claims

that interpreting the saddle points requires both ray and wave theoretic con-

cepts, which commits us to the ray theoretic concepts, but not some meta-

physical commitment to rays in a way he claims that Batterman’s position

requires, in terms of some kind of emergence. Thus the CAM approach, the

14Nussenzveig (1992, pg. 89-90); Adam (2002, pg. 282); Grandy Jr (2005, pg. 54).15Nussenzveig (1992, pg. 94, 102-107); Adam (2002, pg. 242-243, 284, 287); Grandy Jr

(2005, pg. 143, 159, 170-181).16Nussenzveig (1992, pg. 50); Grandy Jr (2005, pg. 39)

155

move to the complex plane and the way saddle points are physical interpreted

(either directly or by proxy) provides a point of contrast between the PMA and

Batterman, which will hopefully allow for a greater understanding of Pincock’s

metaphysical agnosticism. I will argue that, although the rainbow does not

involve a case of a singular perturbation, the moves involved are very similar

and so can be used to gain information on metaphysical agnosticism.

I will now turn to establishing how the IC and PMA can accommodate faithful

epistemic representation in detail.

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6 | The Inferential Conception and

the Rainbow

6.1 Introduction

In Chapter 4 I argued in favour of understanding the Inferential Conception

and Pincock’s Mapping Account as accounts of faithful epistemic representa-

tion. I also argued that a key to the success of these accounts would be how

they accommodate idealisations. In this chapter and the next I will explore

how the two accounts aim to do this, focusing on two issues. The first issue

is the question (2.b), ‘how do the putative accounts of faithful epistemic rep-

resentation account for the relationship between faithfulness and usefulness?’

Answering question (2.b) will involve answering the following type of ques-

tions: what do the accounts have to say about how these two features are

related, if anything? Can they provide any explanation for how a less faith-

ful representation can be more useful than a more faithful one? And if so, is

this an explanation that can be generalised to all types of idealisations,1 or do

1The issue of how many ‘types’ of idealisation is an open and active one in the literature.McMullin (1985) offers a defence of Galilean idealisation, while Jones (2005) argues thatsome of these sorts of idealisations would be better accommodated as abstractions underhis scheme. Weisberg (2013) offers a different way of classifying idealisations, according towhich he distinguishes three types of idealisation. Rohwer & Rice (2013) offer another typeof idealisation consistent with Weisberg’s scheme and raise the possibility of there beingeven more. They also, correctly, criticise Weisberg’s classifying the idealisations focused onby Batterman as ‘minimalist’ idealisations (ones that isolate a single cause); they cannotbe such idealisations as Batterman explicitly denies he is identifying a causal feature of thesystems under investigation (Batterman 2001, §8.4). In this discussion I will be focusing

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the differences between Galilean idealisations and singular limit idealisations

prevent the explanation from being generalised? The second issue concerns

the structural resources used by the accounts to accommodate the idealisa-

tions. Are the same structural resources used to accommodate different types

of idealisations?

I will find problems with both accounts in my investigation of these issues.

In this chapter I will be focusing on the IC, and I will look at the PMA in

the next chapter. The IC’s use of the partial structures framework is initially

promising. I will argue that the partial relations between partial structures

and the notion of partial truth are well suited to accommodating the variable

notion of faithfulness, and its relationship to the usefulness of a representa-

tion. However, I will identify a lack of consistency of how the R3 component is

employed across examples that are supposedly explanatory of how the partial

structures framework functions. This will lead me to identify a problem with

how to understand surplus structure on the IC. In attempting to answer this

question, I will identify four different types of surplus structure. I will argue

that these can be restricted to different parts of the SV and IC. This restric-

tion will result in a different way in which the partial structures framework

is to be employed in scientific representation in general2 and to how I believe

it has to be accommodated when the partial structures framework is used in

mathematical representation.3 Although this proposal looks promising, I con-

on Galilean and singular limit idealisations as they pose the most interesting philosophicalquestions.

2By ‘in general’ I mean instances where the representational vehicles involve interpretedmathematics. This instance of scientific representation will be contrasted to the instanceswhere we are concerned with how uninterpreted mathematics can represent the world. In thegeneral scientific instances, the introduction of surplus structure is prima facíe understoodas a move to uninterpreted mathematics. It is the lack of interpretation of the mathematicsthat will generate the problem for understanding surplus structure on the IC, though thesituation is move complicated than a simple more to uninterpreted mathematics, as outlinedin footnote 3.

3What constitutes surplus structure is more involved than the brief comment in the pre-vious footnote conveys. It requires “ ‘new’ structure which is genuinely ‘surplus’ and whichsupplies new mathematical resources” (French 1999, pg. 188). French characterises this asthe embedding of the original structure into a family of further structures, an insight at-

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struct a dilemma for one type of surplus structure that the IC should be able

to accommodate. This dilemma needs to be answered for the IC to be an

acceptable account of mathematical representation; I outline several possible

solutions.

To finish my discussion of the problems of the IC and the PMA I will return

to the features of the rainbow I identified in the previous chapter: the singular

limit of taking the size parameter to infinity, and the introduction of complex

numbers in the CAM approach. In this chapter, I will discuss these features as

possible examples of the introduction of surplus structure. They will provide

a test case for the solutions I outline in response to problems raised in this

chapter.

6.2 The Inferential Conception and Idealisation

The Inferential Conception’s account of idealisation rests upon the partial

structures framework. It employs both aspects of this framework: the notion

of partial structures (and relations) and the notion of partial truth. I will

address how the notion of partial truth should be employed by the IC before

turning to the notion of partial structures. I argue that the notion of partial

truth should only be employed at the end of an application of the IC, after

the interpretation step. This raises the question of whether partial structures

should be employed during the derivation step of the IC, which subsequently

raises an issue for how to properly understand the notion of surplus structure.

tributed to Octávio Bueno. This move from a structure to a family of further structures doesnot change the point, however, that the structures must still be uninterpreted mathematics.This is made clear through French’s later discussion of how group theory provides us withmore structure that is ruled out by physical laws (French 1999, pg. 195) and the discussionof the relationship between group theory and quantum mechanics. Further evidence is givenin Bueno and French’s discussion of surplus structure: “we typically have surplus structureat the mathematical level, so only some structure is brought from mathematics to physics”(Bueno & French 2012, pg. 88, second emphasis mine).

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6.2.1 Partial Truth and Idealisation

Proponents of the Semantic View (SV) of theories who employ the partial

structures framework support a view of idealisation that involves ‘as if’ de-

scription, where idealisations are taken as if they were true. This view of

idealisation can be underpinned by the notion of partial truth.4 The idea

is that an idealisation, although strictly false, can be considered true within

a restricted domain, and this is exactly what the notion of partial truth is

supposed to capture (da Costa & French 2003, pg. 19):

[W]e say that S is pragmatically true in the structure A if there exists an

A-normal B in which S is true, in the correspondence sense. If § is not

pragmatically true inA according to B, then § is said to be pragmatically

false in A according to B.

This view of idealisation is readily applicable to both idealised theories and

terms, which can be thought of as “idealizing descriptions laid down within

a theoretical context” (da Costa & French 2003, pg. 163). For example, to

“describe an electron as if it were a point particle is to lay down a bundle

of properties that have meaning only within a model or, more generally, a

structure”.

Thus, provided that all idealisations can be understood as ‘as if’ descrip-

tions, then partial truth can be used to explain why an idealisation might be

useful despite being partially faithful. Provided that the context within which

the idealisation is being used allows the idealisation to be considered ‘as if’

it were true, i.e. as partially true, then we have an explanation for why the

idealisation is useful: within the context the idealisation is true.

Further, the ‘partiality’ of the truth gives a measure of how faithful the

representation is. However, this must be measured external to the theoretical4See: French & Ladyman (1998), pg. 56-58; da Costa & French (2003), pg. 163; and

§4.2.1 - The Partial Structures Framework.

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context, in that the representation must be compared to other representa-

tions to gage how partially true it is. Thus we can see how faithfulness and

usefulness are related, though come apart: one is external to the theoretical

context of the idealisation (the faithfulness, how partially true the idealisa-

tion is compared to the target) and the other is internal to the context (the

usefulness, that the idealisation is partially true within the context).5 The

threat to this view comes from idealisations which we cannot consider as ‘as

if’ descriptions. Galilean idealisations can be straightforwardly considered as

‘as if’ descriptions. It is not clear that the same can be said for singular limit

idealisations.

Galilean idealisations were first introduced by McMullin (1985).6 They

can be generally characterised as “a deliberate simplifying of something com-

plicated (a situation, a concept, etc.) with a view to achieving at least a partial

understanding of that thing” (McMullin 1985, pg. 248) and involves “assuming

as true . . . a proposition that is false” (McMullin 1985, pg. 255). McMullin

goes on to identify two places in which these simplifications can occur, either

the conceptual representation of the target (i.e. the representational vehicle,

the model) or the “problem situation” itself, and two different ways in which

these simplifications can occur in each place. Altering the conceptual repre-

sentation results in construct idealisation. This alteration can either occur

due to features known to be relevant to the situation being represented by the

model being simplified or omitted, which would be a case of formal idealisa-

5This distinction is how partial truth is used when discussing theories. Partial truthis related to the ‘intrinsic’ characterisation of theories, whereas the ‘extrinsic’ perspectiveconcerns the relationship between theories themselves, and between theories and the world.Strictly, the models cannot be considered partially true from the extrinsic perspective. WhenI said the partiality of the truth of the representations could be compared, I meant that thesize of the Ri could be compared. I will sketch this idea in the next section. See da Costa& French (2003) and French & Saatsi (2006) for more on the extrinsic/intrinsic perspectivedistinction.

6I will be using the general characterisation of Galilean idealisation below. Critical devel-opments and other characterisations of Galilean idealisation can be found in the referenceslisted in footnote 1.

161

tion, or through irrelevant features being left unspecified, which would be a

case of material idealisation. Altering the problem situation results in casual

idealisation. This involves the “move from the complexity of nature to the

specially contrived order of the experiments” and can occur either literally or

through thought experiments (McMullin 1985, pg. 365). These idealisations

are justified pragmatically. This is borne out in the idea of deidealisation,

the “adding back” of theoretical corrections which cannot be ad hoc if they

are to be justified deidealisations (McMullin 1985, pg. 259). The idea is that

Galilean idealisation should be capable of being deidealised given sufficient

computational power and sophisticated mathematical techniques.7

Singular limits are limits that result in a mathematical discontinuity, an

asymptote. They are the main focus of the work of Batterman.8 The most

important difference between idealisations due to singular limits and Galilean

idealisations, at least as argued for by Batterman, is that the singular limit

is essential to the idealisation. One cannot represent the situation properly

if one attempts to deidealise by stepping back from the limit. There is much

debate over the status one should grant to the models that result from taking

a singular limit. Batterman holds them to play essential explanatory roles9

whereas proponents of the partial structures approach often claim that the

resulting structures are cases of surplus structure10 (Bueno & French 2012, pg.

91). It is this claim that the structures which result from singular limits to be

surplus structures that causes a problem for the IC. But to see this problem,

we must take a step back.

In the above introduction of the ‘as if’ view of idealisations I mentioned

7See Weisberg (2013).8See Batterman (2001, 2005b, 2008, 2010) amongst others.9More precisely, the asymptotic limits produce “structures and properties [which] play

essential explanatory roles” (Batterman 2001, pg. 21).10They also argue that in the absence of an appropriate account of explanation the claim

that such structures do play an indispensable explanatory role cannot be sustained (Bueno& French 2012, pg. 101-103).

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that theoretical terms could be understood as descriptions “laid down within

a theoretical context” (da Costa & French 2003, pg. 163, my emphasis). When

discussing idealisation in the context of scientific representation in general, as

is typical for discussions of how the partial structures facilitates the SV, the

theoretical context is provided by the interpretation of the model. For example,

take the idealisation of a cannon ball as only being under the influence of

gravity in a typical projectile problem in Newtonian Mechanics. The equations

used are interpreted in terms of Newtonian Mechanics. F , m, a and so on have

the interpretation of Newtonian force, mass and acceleration. When we are

mapping to an uninterpreted mathematical structure, however, there is no

theoretical context.

In fact, this lack of theoretical context is one of the benefits of moving

to surplus structures: it allows us to identify structural features, or relations

within the lower model that we could not without the extra mathematical

resources, as in the case of the relationship between quantum mechanics and

group theory (French 1999, §3). Group theory is useful for describing the

symmetry of systems. By understanding quantum states as displaying certain

symmetries, group theory can be introduced in order to give a more funda-

mental understanding of the symmetries. “An atom or an ion, whose nucleus

is considered as a fixed center of force O, possesses two kinds of symmetry

properties: (1) the laws governing it are spherically symmetric, i.e., invariant

under arbitrary rotation about O; (2) it is invariant under permutation of its f

electrons” (Weyl 1968, pg. 268), quoted in (French 1999, pg. 194-195). French

explains that the symmetries of type (1) are described by the rotation group,

while those of type (2) are described by the finite symmetric group of all f !

permutations of f things.

With symmetries of type (2) we find that the appropriate system space

for two indistinguishable individuals is reducible into two independent sub-

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spaces, which are symmetric or anti-symmetric (French 1999, pg. 197). We

have surplus structure both from the ‘mix-symmetry’ state functions that de-

scribe the parastatistics, and if the state is one of bosons, then the fermion

subspaces can be considered surplus structure and vice versa (French 1999, pg.

198). Whether the quantum state is of bosons or fermions, however, requires

a map back to the empirical set up to establish which subspace is empirically

adequate. i.e. we do not establish what structure is surplus until we have

a theoretical context. Thus, a move to surplus structure cannot strictly be

called an idealisation, at least while we are within the mathematical domain,

as there is no theoretical context to provide the ‘as if’ description.11

Being unable to call a move to surplus structure an idealisation until we

move back to an interpreted model does not seem to be problematic when

partial structures are used for the Semantic View. However, it does seem

problematic when using partial structures to account for the applicability of

mathematics via the IC. According to the IC, there are only two places where

the mathematics has an interpretation: the initial and the final empirical set

ups. The immersion mapping takes us into the realm of uninterpreted math-

ematics, and the immersion step takes us back to an interpreted (in terms of

the world, or scientific context) structure. This means that during the ap-

plication of the IC, we cannot speak of there being an idealisation once we

have performed the immersion mapping, until we perform the interpretation

mapping to an empirical set up. So strictly on the IC, the only time we have

an idealisation is once we have obtained the final empirical set up, and this,

the interpreted (partial) structure of the empirical set up, is what we should

refer to when talking of an idealisation or idealised model on the IC.

This raises two further problems. First, strictly speaking, we do not have

11In the case of (1), there are many more representations of the rotation group thanphysically occur; hence we have surplus structure from group theory (French 1999, pg. 195).However, identifying what structure is surplus and what is physically significant requiresreinterpreting the mathematics, that is, providing a theoretical context for it.

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idealised representations on the IC. Our representational vehicles are mathe-

matical structures and the targets are the empirical set ups. The IC accounts

for how mathematics can be used to obtain surrogative inferences about em-

pirical set ups. How, then, do we have idealised representations of the world?

The answer to this is to recognise that the empirical set ups are the represen-

tational vehicles, i.e. the models, of the SV. As such they can be considered

idealised representations when they are acting as vehicles for the SV, when it

provides an account of how scientific models represent the world.

The second problem is whether surplus structure makes sense within the

IC. Bueno’s and French’s response to Batterman’s position on singular limits

is to iterate the immersion mapping and consider the second mathematical

model to be surplus structure.12 This response is not possible if surplus struc-

ture is the move from an interpreted mathematical model to uninterpreted

mathematics, as the first immersion mapping has already taken us to an unin-

terpreted mathematical model. I will discuss this problem further below, after

looking at different roles partial structures have in the SV and the IC.

6.2.2 Partial Structures and Idealisations

Partial truth requires partial structures, but partial structures can provide

more than partial truth. They are employed in solutions to problems in major

areas in the philosophy of science, such as the structure of theories, inter-theory

relationships, the theory-data-phenomena relationship, and of course the ap-

plicability of mathematics. One can see that there are many more motivations

for adopting the partial structure framework than the associated notion of

partial truth.

One such motivation is to ground the notion of faithfulness. Above I argued

that partial truth could be used to explain how the partial faithfulness and

12Bueno & French (2012) provide this response to Batterman (2010).

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usefulness of representations could be related. This explanation was vague,

talking of ‘how partially true’ an idealisation is as reflecting the faithfulness of

the representation. This notion can be made more precise by looking at the

(partial) relations between the (partial) structures that describe the idealised

model and the empirical set up, world, or whatever we are judging the faith-

fulness of the model against (i.e. the target). The faithfulness of the idealised

model can be measured by comparing the Ri of the model to the Ri of our

target, in the same way the ‘degree’ of similarity or approximation of models

can be measured (da Costa & French 2003, pg. 50-51), (da Costa & French

1990, pg. 261).

We can further supplement the explanation of why idealised models are

useful with the partial structures framework. In addition to the theoretical

context setting down what we are happy to consider ‘as if’ it were true, the

relationship between the target and the idealised model (a relationship that

might compose numerous partial structures and partial mappings) allows us to

see how that idealised model has some link to the target, yet be strictly false

when compared to the target directly. Thus the partial structures framework

provides an explanation, and a formal grounding for the explanation, of why

partially faithful models can be extremely useful.

As well as discussing the role that partial structures can play as a whole,

we can identify the roles that the individual components can play:

R1 Similarities between models, or between the empirical set up and the

mathematical structures.

R2 Dissimilarities between models, or between the empirical set up and the

mathematical structures.

R3 Providing the ‘openness’ of structure which allows for the introduction

of group theory as surplus structure (Bueno et al. 2002, pg. 514); “mis-

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matches” between the empirical set up and the mathematical structure

when using the partial structures as a framework for accommodating

idealisations (Bueno & Colyvan 2011, pg. 359); and housing the mathe-

matical “technique” of the renormalization group (Bueno & French 2012,

pg. 103).

The descriptions of what the R3 component can be used for appear to be

confused. The first two roles indicate that the ‘openness’ of a partial structure

comes from the R3 component, but this ‘openness’ is cashed out in two very

different ways.

In the group theory case, this ‘openness’ involves the introduction of gen-

uinely new structure, i.e. surplus structure. Here group theory is appealed

to as a way of providing a greater understanding of the Bose-Einstein statis-

tics initially used to describe superfluidity. In attempting to provide a model

for the observed behaviour of liquid helium, London drew an analogy with a

Bose-Einstein ideal gas. Bueno et al. argue that it is the structural similarity

between the λ-point of helium (the phase transition that liquid helium under-

goes at 2.19°K) represented graphically and the discontinuity in the derivative

of the specific heat associated with a Bose-Einstein condensation that provides

this analogy, and the introduction (at a higher ‘level’) of the Bose-Einstein

statistics (Bueno et al. 2002, pg. 512). What is important for the present

discussion is that they claim the introduction of group theory allows for an

understanding of the Bose-Einstein statistics via considerations of symmetry,

the symmetry being represented group-theoretically (Bueno et al. 2002, pg.

514). The particular features of group theory that are appealed to are the ex-

istence of further “bridges” within the mathematics of group theory, specifically

the reciprocity relationship between the permutation group and the group of

all homogenous linear transformations (Bueno et al. 2002, pg. 508) (French

1999, pg. 199). The idea here, then, is that the ‘openness’ involves the use

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of mathematics to gain a greater understanding of the structural relationships

within the original model by moving to a domain where we can explore the

structural relationships between the uninterpreted mathematics more easily

than we can between the interpreted mathematics, the moves being restricted

or obscured by the physical interpretation.

However, in the framework case, the ‘openness’ is cashed out in terms of

mismatches between the target and the (‘idealised’) mathematical structure

acting as the representational vehicle (Bueno & Colyvan 2011, pg. 359):

The R1 and R2 components of the partial relations in the empirical set

up are mapped via the relevant partial isomorphism or partial homomor-

phism into the corresponding partial relations in the mathematical struc-

ture. However, the R3 components are left open. These components

correspond to the features of the idealization that bring mismatches be-

tween the actual empirical world and the mathematical model.

This quotation seems to indicate that all idealisations should be located in

the R3 component of the partial structure. Yet this component is supposed

to include statements for which we do not know whether they hold in the

structure.13 This causes a problem, as we can see if we pay attention to the

details of the economics example Bueno and Colyvan use to motivate their

claims that we can use the partial structures approach as a framework for

idealisations. This example is the attempt to model the rationality of agents

involved in running a business, where the goal is to maximise profits. One of

the idealisations involved is the ‘quantity idealisation’: the quantity produced,

qs, is assumed to equal the quantity demanded, qd (which is a function of price:

qd = D(p)): qs = qd = q. This allows us to calculate the profit as the difference

between the gross receipt, R, and the cost of production, C.14 i.e. Profit =13See the introduction of the partial structures framework in §4.2.1 - The Partial Struc-

tures Framework, and da Costa & French (2003).14The gross receipt is the product of the price and quantity demanded R = pqd. The cost

of production is a function of the quantity produced: C = C(qs).

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R − C = pq = C(q).15 As the quantity idealisation is an explicit statement

about the model, it appears to be a statement that we do know is satisfied,

and so should be placed in R1. This immediately contradicts the claim in

the quotation that such idealisations should be placed in the R3 component.

However, we explicitly make this statement as an idealisation, and so know that

it is false, or at least some kind of misrepresentation, and so should be placed

in the R2 component. But if it is placed in the R2 component, then one cannot

use the idealisation to say anything positive about the world. The same goes

for a variety of other idealisations: frictionless planes, point particles, etc. It

appears there is a fatal flaw in the partial structures’ approach to idealisations,

due to the contradictory epistemic attitudes one is supposed to take. We do not

know which of the Ri components should contain the idealisation statement. I

shall call this problem the partial structures’ epistemic problem for idealisation.

There is a possible solution to the epistemic problem.16 One must remem-

ber that the content of representations on the IC is supposed to be uninter-

preted mathematics. Thus, when one equates qd and qs, one is not making

any statements about the world. One is simply equating two mathematical

variables to, for example, help ease the tractability of the problem one is at-

tempting to solve. Thus one is not obliged to place the idealised statement into

R2. But where should one place it? This idealisation does not seem capable of

being understood as being open; either qd = qs or qd ≠ qs. Thus I would argue

that it should be placed into the R1 component. So even with this possible,

and I think correct, solution to the epistemic problem, the claim that one can

use the R3 component to capture the openness of idealisations looks to be false.

Further, as the introduction to the partial structures shows, the matches and

mismatches between the target system and an idealised model are found in

15The price and quantity at which profits are maximised can now be calculated throughsimple analysis: pmax when dR−C

dq= 0.

16Thanks must go to Octávio Bueno for providing this response to the problem in con-versation.

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the R1 and R2 components respectively in order for the ‘as if’ understanding

of idealisation to succeed.

A charitable interpretation of the openness framework claim might be that

features which are ‘screened off’ through idealisation come to be found in the

R3 component, such as in the case of the ideal gas. Here, an idealisation results

in the omission of some information, namely the internal structure of the gas

molecules (Jones 2005, pg. 189-190) (McMullin 1985, pg. 258). McMullin’s

explanation of these idealisations as ‘material’ idealisations in some Aristote-

lean sense is amenable to this charitable reading. He talks of the model as

including unspecified parameters or parts, of them being “open for question in

a different context”, that such models are “necessarily incomplete; they do not

explicitly specify more than they have to for the immediate purposes at hand”

(McMullin 1985, pg. 262-263). The charitable reading, then, might be that

the idea of ‘openness’ that Bueno and Colyvan are appealing to in the context

of the economics example is that such classical economics models ‘cover’ cer-

tain features of the world, in terms of leaving them unspecified, which can be

specified in a different context. The R3 component seems suitable for this job,

as it is supposed to accommodate statements for which we do not know, or for

which it has not yet been established, whether they hold in the structure or

not.

I do not think that this charitable reading is successful. It fails because this

‘openness’ in terms of mismatches between target and vehicle is given in the

context of a framework that is supposed to be general and accommodate all

types of idealisation. The charitable reading relies on one type of Galilean ide-

alisation. This would tie the ‘openness’ brought about by mismatches between

target and vehicle to this single type of Galilean idealisation for the partial

structures framework. Thus the framework would not be able to handle any

of the other numerous types of idealisation. The charitable reading should

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therefore be dispensed with as a way of saving this ‘openness’ claim in general.

If all Bueno and Colyvan meant by this claim is that the economics example

is a case of material Galilean idealisation, then this notion of ‘openness’ can

survive.

The third role given above for the R3 component is that it houses the

‘technique’ of the Renormalization Group (RG).17 I will ignore any possible

significance over the terminology used to describe the RG. I will instead focus

on the fact that an application of the RG involves taking singular limits. This

allows one to understand such an application as a singular limit idealisation,

and therefore might involve surplus structure. This should mean that this

role of the R3 is the same as the role for group theory. However, it is not

clear that this is the case as the group theory role is within the context of an

SV use of the partial structures framework, whereas the RG role is discussed

within the context of an IC use of the partial structures framework. The

RG role implies the introduction of surplus structure due to a move from

an uninterpreted mathematical model to another uninterpreted mathematical

model. The London model group theory example introduces surplus structure

due to a move from an interpreted mathematical model to an uninterpreted

one, as well as implying surplus structure is related to the analogies drawn

between the Bose-Einstein and group theoretic models. There is clearly a

difference in how to accommodate surplus structures between the SV and the

IC.

The above discussion shows that there is some motivation to adopting

partial structures in the derivation step to accommodate surplus structure.

But this motivation requires refinement, due to the difficulty in understanding

exactly what surplus structure is on the IC. I now turn to clarifying the notion

17See Batterman (2005a, 2011) for some of Batterman’s introductions and discussionsof the renormalisation group, and Bueno & French (2011) for an explanation and responsefrom the IC perspective. Kadanoff (2000) provides a textbook introduction to the topic.

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of surplus structure, identifying what role the IC should accommodate.

6.2.3 Surplus Structure on the Inferential Conception

Thus far I have been working with a definition of surplus structure as the in-

troduction of genuinely new structure, but I have also talked about it as the

move from interpreted mathematics to uninterpreted mathematics. That is,

the genuinely new structure is always new uninterpreted mathematical struc-

ture. However, there is an ambiguity in the literature over what constitutes

surplus structure. The notion of surplus structure as the introduction of new

structure and involving a move from interpreted to uninterpreted mathematics

is based upon the way the term is used in the discussion of how group theory

is related to Bose-Einstein statistics in the London-London case by Bueno et

al in their (2002). This example is of the same kind as the case of analytic

S-matrix theory in elementary particle physics (Redhead 1975, pg. 88) (Red-

head 2001, pg. 80). Here scattering amplitudes are considered as functions of

real-valued energy and momentum transfer. These functions were continued

into the complex plane and axioms concerning the singularities of the func-

tions in the complex plane were introduced to set up systems of equations

which “controlled” the behaviour of the scattering amplitudes considered as

functions of the real, physical, variables. The common feature of these two

examples, group theory and S-matrix theory, is that “there was no question of

identifying any physical correlate with the surplus structure” (Redhead 2001,

pg. 80).18 These two examples are contrasted to cases where “what starts as

surplus structure may come to be seen as invested with physical reality”, that

is, what starts as surplus structure, uninterpreted mathematics, gains an in-

terpretation. Redhead offers the examples of the kinetic theory of matter as

viewed by positivists before the discovery of Brownian motion (Redhead 1975,

18The symmetries of type (1) in the group theory and quantum mechanics example fromFrench (1999) is this kind of surplus structure.

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pg. 88), energy (Redhead 2001, pg. 80-81) and the negative energy solutions

to the Dirac equation (Redhead 2001, pg. 83).19 These were first thought to

be artefacts of the mathematics, then came to be interpreted as ‘holes’ in a sea

of negative energy electrons before finally being reinterpreted as positrons.20

Redhead also outlines two further roles for surplus structure: certain ideal-

isations, such as representing physical magnitudes by the real numbers, can

“[add] (surplus) structure to the physics rather than stripping it away, as in

alternative senses of idealization”, i.e. formal Galilean idealisations (Redhead

2001, pg. 83); and as a way of accommodating Hesse’s account of how theories

develop through analogous models.2122

The last of these roles for surplus structure, that of accommodating Hesse’s

account of theory development, is clearly not a role that the IC should accom-

modate: this is something that the SV handles as it involves inter-theory

relations (da Costa & French 2003, pg. 47-52). The second and third roles can

be accommodated by the IC. The notion of there being more mathematical

structure compared to physical structure, what underlies the particular type

of idealisation that Redhead highlights as the third role, is noted by Bueno

and Colyvan in their criticism of Pincock’s original mapping account (Bueno &

19The symmetries of type (2) in the group theory and quantum mechanics example fromFrench (1999) is this kind of surplus structure.

20I outlined Dirac’s attempts at interpreting the negative energy solutions in §2.2.2.21Redhead, summarising Hesse (1963) states that “theories can develop by providing

physical interpretation for [the] surplus structure and [she] argues persuasively that thejustification for obtaining a successfully strongly predictive theory by this manoeuvre de-pends on pre-theoretic material analogies with an analogue model which already providesan interpretation for the surplus structure” (Redhead 1980, pg. 149).

22Teller provides an alternative distinction between two types of surplus structure:strongly surplus structure where the mathematical formalism has interpreted elements thatnever have actual physical correlates, and weakly surplus structure where the mathematicalformalism contains some uninterpreted elements (Teller 1997, pg. 25-26). He also describesthe example of the negative energy solutions of the Dirac equation as being apparent surplusstructure. Teller is clearly working with a conception of surplus structure similar to the oneI have been thus far, that surplus structure has to be uninterpreted mathematics. However,Teller also goes on to argue that genuine surplus structure should be “shunned”, both inthe weak and strong cases, which leads me to think that his conception is slightly differentto mine. I will not discuss Teller further, as I will argue that the Redhead conceptions ofsurplus structure can be accommodated by the IC and the SV.

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Colyvan 2011, pg. 348). However, given what I have argued above about ideal-

isation, this ‘surplus structure’ should only present itself in the final empirical

set up, after mapping from the resultant structure (as within the derivation

step, before the interpretation mapping, we haven’t finished representing any-

thing yet). The second case can be accommodated by the IC in a similar way.

Bueno and Colyvan explain that the multiple interpretations of the Dirac neg-

ative energy solutions consist in adopting different partial morphisms from the

resultant structure of the Dirac equation, to different empirical set ups (Bueno

& Colyvan 2011, pg. 364-366). Here the surplus structure would be mapped

to the Ri component of the empirical set up depending on the interpretation,

e.g. the R2 component when they are dismissed as being unphysical.

The crucial distinction between these two cases and the first is at which

stage the surplus structure is identified within the IC. The second and third

cases concern structure that is surplus after the derivation step, and involve

the relationship between the resultant structure and the empirical set up. As

the first role involves mathematics that will not be physically interpreted, this

suggests that the surplus structure will be structure obtained after the iteration

of the immersion mapping, i.e. it will be in the Model 2 of the iterated IC.

The problem I intend raise concerning the plausibility of surplus structure on

the IC is targeted at the first type of surplus structure. ‘Surplus structure’

will refer exclusively to this first role from this point on.

While there is a clear division between interpreted mathematics (the scien-

tific models) and the uninterpreted mathematics that constitutes the surplus

structure when discussing the SV, this division completely breaks down when

one applies the IC. The immersion and interpretation step can be iterated;

Bueno and French argue that this iteration is the way to account for cases of

surplus structure, such as an application of the RG (Bueno & French 2012, pg.

91-92):

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One of the features of the account we advocate is that it can accom-

modate the role of surplus mathematical structure, whereby a given

physical structure can be related via partial homomorphisms (or some

other partial morphism) to a suitable mathematical structure, which in

turn is related to further mathematical structure, some of which can

then in turn be interpreted physically (see Bueno (1997), Bueno et al.

(2002) and Bueno & Colyvan (2011)). We can represent such a surplus

structure within the inferential conception by straightforward iteration:

the initial mathematical model (Model 1) that is used to represent the

original empirical set up is itself immersed into another model (Model 2),

which gives us the surplus structure, and the results are then interpreted

back into Model 1, which only then is interpreted into the physical set

up.

I have challenged that such an iteration can lead to surplus structure. Model

1 is already uninterpreted mathematics, and so ‘surplus’ in the sense of being

uninterpreted. An iterated immersion mapping to Model 2 does not alter this.

There must be more to the notion of surplus structure. When discussing the

application of group theory (both to quantum mechanics and superfluidity)

it is the shift from interpreted to uninterpreted mathematics in addition to

that uninterpreted mathematics providing genuinely new structure which does

the novel work (Bueno et al. 2002, pg. 508-509). This still does not solve the

problem; there must still be yet more to the notion of surplus structure.

Another feature of surplus structure can be identified by looking to how

the IC can accommodate the application of mathematics to other mathematics.

The situation is the same as the iteration case: one maps from uninterpreted

mathematics to other uninterpreted mathematics. Bueno and Colyvan lever-

age the IC to explain how the unification of the two different domains of the

complex numbers and the theory of differential equations can be understood in

terms of importing structure into different domains (Bueno & Colyvan 2011,

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pg. 364). What is emphasised is the introduction of “new inferential relations

[where] [u]nification emerges . . . as the result of establishing such inferen-

tial relations among apparently unrelated things”. So the reason that surplus

structure is so useful is three fold: there are surplus structural resources; un-

interpreted mathematics provides such surplus structural resources; and these

surplus uninterpreted mathematical resources provide new inferential relations

to bring to bear on our original structures. Described as such, there seems to

be less problem in moving from one mathematical structure to another, and

describing the second as surplus structure.

There is a problem, however, in how this move is supposed to be accommo-

dated if one takes surplus structure to involve partial structures. In the group

theory case, it is argued by Bueno et al. that group theory comes to enter the

discussion through the openness of London’s ‘rough and preliminary’ model,

that is, through the R3 component: “Retaining the component associated with

Bose-Einstein statistics, understood group-theoretically, new elements were in-

troduced . . . corresponding to a move from our R3” (Bueno et al. 2002, pg.

514). From this discussion I think one has to infer that group theory is some

how ‘in’ the R3 component already, for it be capable of being introduced. Or

perhaps more coherently, there are structures in the R3 that are (partially)

iso-, homo-, or otherwise, -morphic to the structures of group theory. But this

begs the question of what limits the R3? If the R3 is to truly represent ‘open-

ness’, it has to be capable of introducing more than one type of mathematics,

more than group theory. It has to be open to the agent to introduce various

different types of mathematics; the physical interpretation is what picks out

group theory (and only certain parts of group theory) as being appropriate

to the application at hand, not the content of R3.23 Given this, one must be

23As we are dealing with the first type of surplus structure here, the physical interpreta-tion comes from mapping the group theoretic results to the Bose-Einstein model, and thisbeing mapped to an empirical set up. The group theoretic resultant structure does not geta direct physical interpretation.

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capable of extending the original structure to any type of mathematics. This

problem should be recognised as being a serious problem, as it appears any

time we apply mathematics. There is no way to predict whether we will need

to turn to surplus structure in any given application of mathematics (or at

least any novel application of mathematics). Thus, any application of mathe-

matics will require all mathematical structures in the R3 (or again, structures

that are (partially) -morphic to all mathematical structures).

It appears that if we understand surplus structure in terms of partial struc-

tures, then our partial structure must contain all mathematical structures in

the R3 component. Call this the Very Large R3 Problem. This problem creates

a dilemma for the proponents of the partial structure framework. Either they

must admit the Very Large R3, or they must introduce some way of constrain-

ing the R3. However, as has been argued, the derivation step takes part in the

domain of uninterpreted mathematics, where physical considerations do not

play any role (to avoid the epistemic problem and to maintain the ‘as if’ view

of idealisation). Thus there is a problem concerning what we can appeal to in

order to constrain the mathematical moves we make (and hence R3). Call this

the Mathematical Restriction Problem. The Mathematical Restriction Prob-

lem can be understood as an attempt to cut off the Very Large R3, by providing

a reason to only include structure that is (partially) morphic to the relevant

mathematics (e.g. group theory) in the R3. If this problem cannot be solved,

then we have to deal with the Very Large R3 Problem. These two problems

thus compose the two horns of a dilemma. Let us start by attempting to deal

with the Mathematical Restriction Problem.

There are two ways in which we can attempt to restrict the mathematics

that might be required for the R3: reasons that are either internal or external

to the mathematics. Looking first at the internal reasons: there might be some

intrinsic feature of the mathematics that leads us to prefer to the adoption of

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a certain transformation than other. For example, we might decide that a

Fourier transform is more appropriate than a Taylor expansion due to the

original function being periodic. This sort of restriction, however, is not the

kind of restriction we require. This only restricts which moves we take, not

which moves are available. This is a ‘use’ restriction, in that it restricts what

mathematics we use in the model. What we are looking for is a restriction

that prevents certain mathematics from being in the R3 in the first place, i.e.

a restriction in principle rather than in practice. A possible solution might

be to opt for a kind of negative solution. Some extensions can be seen to be

very ‘natural’, in that there are strong internal mathematical reasons for the

extensions. Take as an example the extension of real functions into the complex

plane. It is reasonable to claim that to properly understand any real function,

one has to extend it into the complex plane. Hence, this extension is ‘natural’

in that it is required for a proper understanding of the function at hand. We

can then put forward the claim that any non-‘natural’ extension should not be

included in the R3 component. While promising, this requires more work to see

whether it is a viable option. It is plausible that this non-‘natural’ restriction

might be too restrictive in some instances.

I think that there are four external reasons to restrict the R3:

(a) physical interpretations;

(b) tractability concerns;

(c) intentions of the scientists/representing agents;

(d) pragmatic, contextual and heuristic reasons.

(a) can be rejected immediately. We are dealing with the first type of surplus

structure, which has no chance of being physically interpreted. Physical inter-

pretations therefore have no bearing on what we need to place in the R3 for

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this type of surplus structure. (b) can be dismissed as tractability can be con-

sidered a ‘use’ restriction rather than an in principle restriction. Tractability

will prevent us from pursuing certain lines of mathematical enquiry, but it will

not prevent them from being possible avenues and hence having to be included

in the R3.24

(c) and (d) come together and are initially the most promising solutions.

The agents performing such representations have intuitions about what sort of

mathematics would be best to find the sort of structural relations or features

that might have a bearing on the models they require the surplus structure

to enlighten. This seems promising: the solution can be incorporated into the

agents intentions, provided that the earlier comments concerning the agents

intentions25 and the idealisation of the IC26 hold. This is only the sketch of a

solution, however. A lot more needs to be said by the proponents of the IC

on exactly what constitutes the heuristic, pragmatic and contextual consider-

ations, given that they are relied on so heavily throughout the discussion of

the IC and other applications of the partial structures framework.

So much for the first horn of the dilemma. Let us assume for the sake of

argument that the above possible solutions fail. What of the second horn of the

dilemma, the Very Large R3 Problem? If we cannot restrict the mathematics

24This dismissal might be too quick. Tractability can be considered both internal andexternal to the mathematics. For example the Mie solution provides an exact solution forthe scattering of light by a water droplet for the rainbow. It converges slowly, but solutionscan be found with modern computers. The mathematics is therefore tractable internallyin the sense that we can solve the equations. One can consider it to be intractable in anexternal sense, however, as physicists claim that it is hard to ‘see the physics’ within theMie solution.

25See §4.2.1, where I argue that the intentions should not be included in the representationrelation, on pain of fixing the target of the representation, and relate to the particularstructural relation involved in a particular representation. I also discussed the role pragmaticand contextual factors played in identifying the empirical set ups and structural relations.The discussion there was focused more on the interpretation mapping. If similar argumentshold for the interpretation mapping, then this response has some plausibility.

26I’m referring here to the second way in which the IC might be idealised, that “the math-ematical formalism often comes accompanied by certain physical interpretations” (Bueno &Colyvan 2011, pg. 354). This was also discussed in §4.2.1. I rejected any interpretations ofthe mathematics in the derivation step, claiming that the role played by these ‘interpreta-tions’ was accommodated by the pragmatic, heuristic, and contextual considerations.

179

we have to include in the R3, it looks like the Very Large R3 Problem is a

bullet the proponents of the partial structures approach will have to bite. But

there are at least two ways in which the proponents might bite it. They might

either reify the partial structures, which would lead to a problem, or insist that

while they are biting the bullet, it is not a problem as the partial structures

are simply a representational device and so a large R3 has no consequence.

The argument for reifying the partial structures is as follows. The size of

and the content of the partial structures is significant, e.g. the size of the

Ri grounds the notion of how ‘faithful’ a representation is. Hence the size

of the R3 cannot be dismissed. As we are using mathematical structures as

our representation vehicle, we are not dealing with a representation of the

structures, but the mathematical structures themselves. We reify the partial

structures when in the mathematical domain. Therefore the argument that a

very large R3 being required to introduce surplus structure is an argument that

we need a mathematical structure that contains all of mathematics in its R3

component. This, however, raises the problems of how we are to make sense

of and deal with such a large mathematical structure. There is serious work

to be done here to justify such a position, especially given the strong support

the next response has, that the partial structures are representational devices.

Insisting that the partial structures are representational devices has strong

textual evidence.27 This response, while simple, raises some immediate prob-

lems. First, what does it mean to have a representation that suffers from the

Very Large R3 Problem? What is the very large R3 representing? If it is rep-

resenting the situation that we require all mathematics in order to understand

surplus structure, the Very Large R3 Problem is not solved, we’ve merely re-

formed it at the level of the mathematics we are representing. Alternatively,

one might claim that the Very Large R3 isn’t representing anything at all, as a

27See the discussion of this in §8.3. See also French (2012) and (Bueno & French 2011,§9.7).

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way to avoid the problem. But this seems to render the whole notion of partial

structures and surplus structure useless. What is the justification for adopting

the partial structures in some areas and not others (such as the Very Large

R3)? This move risks being ad hoc, and so does not seem like a viable option.

There is clearly work to be done here to explain exactly what consequences

the position of the partial structures being simply representational devices has

on the Very Large R3 Problem. The consequences are not obvious, and the

proponents of the partial structures program owe us more.

A possible way out of this problem is to avoid biting the bullet altogether,

by not requiring partial structures in the derivation step. In order to establish

whether we do, we need to look at whether we can move from one uninterpreted

mathematical structure to another in a way which introduces the necessary

surplus mathematical resources with full structures. It appears that the usual

model-theoretic notion of structure extension is incapable of doing this (French

1999, pg. 188):

However, the mathematical surplus which, it is claimed, drives these

heuristic developments cannot be captured by the usual model-theoretic

notion of a structure extension since this simply involves the addition

of new elements to the relevant domain with the concomitant new rela-

tions. What is required is ‘new’ structure which is genuinely ‘surplus’

and which supplies new mathematical resources. Thus an appropriate

characterisation would be one in which T ′ is itself embedded in an en-

tire family of further structures - T ′′, T ′′′ and so on - and this may then

capture the heuristic role of mathematics in theory construction.:’

Understanding a move to surplus structure as involving a move to a fam-

ily of structures does not require the notion of partial structures or partial

-morphisms. They are introduced by French to accommodate the “incomplete-

ness and openness of the heuristic situation” of applying mathematics to the

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world (French 1999, pg. 192). However, we do need to use partial structures

in the derivation step in order to make use of the partial mappings in the

immersion and interpretation steps.

We can motivate the idea of surplus structure involving an embedding to a

family of structures to solve the problem of whether surplus structure makes

sense on the IC. It seemed that since the defining features of surplus structure

were the uninterpreted mathematics allowing new inferential relations to be

drawn, that there was no difference between the the Model 1 and Model 2

of the iterative application of the IC that one would expect in order to call

Model 2 a case of surplus structure. Now, we can argue that Model 2 isn’t

a ‘model’, but rather a family of models. That is, an iterative application of

the IC that leads to surplus structure moves one from a singular mathematical

model, Model 1, to a family of mathematical models (structures), which is

labeled Model 2.

The question now is whether these solutions to the problems above actually

work. To test this, I will turn to my case study of the rainbow, to the two

instances of what could be considered surplus structure: the taking of the size

parameter, β, to infinity; and the introduction of the complex plane in the

CAM approach. These are two different ways of introducing surplus structure;

for the solutions above to be useful, they have to accommodate both of these

ways of introducing surplus structure.

6.3 The IC Applied to the Rainbow

6.3.1 β →∞

Taking the size parameter to infinity is proposed as a way of obtaining a ray

theoretic representation from wave theory. This is to allow an explanation of

the colour distribution of the rainbow via the wave theoretic concepts that un-

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derpin this ray theoretic representation. Specifically it allows one to appeal to

various values of the wave number, k, to give a different value of the refractive

index, n, for each colour of the rainbow. This is in contrast to obtaining a ray

representation from wave theory by taking the wave number to infinity. This

approach fails if we wish to investigate the rainbow, however, as at the rainbow

angle we have infinite intensity and amplitude. We lose the information we

need to discuss the distinct colours and their particular refractive index, n.

Thus we turn to a ray theoretic model obtained form the Mie solution.

The question that needs to be answered in this section is whether the β

limit is a singular limit, and if it is a singular limit, does the model that

results from it fulfil the definition of being surplus structure, i.e. a family

of uninterpreted mathematical structures that provide additional inferential

relations? Proponents of the IC hold the view that the models that result

from singular limits are ones of surplus structure (Bueno & French 2012, pg.

91). Thus, if the β limit is singular and does not result in surplus structure,

an adjustment will have to be made to the IC.

Grandy Jr states explicitly that we need to take the limit in order to intro-

duce a cut off and convert the infinite sum of the scattering functions into an

integral (Grandy Jr 2005, pg. 120). Here an attempted separation of the ef-

fects of diffraction and reflection is made. The partial wave coefficients, an and

bn (given in (A.121) and (A.122) in the Appendix, and (3.140) in Grandy Jr

(2005, pg. 101), describe the effects of diffraction and the scattering func-

tions SEn and SMn (given in (A.123) and (A.124) in the appendix and (3.141) in

Grandy Jr) describe the reflection. A cut off is introduced by letting an = bn = 12

for n + 12 < β, and 0 otherwise. The conversion of the sum in the scattering

functions to an integral depends on fixing cos(θ) ≡ (n + 12) θ as n → ∞. This

allows us to adopt approximations for πn and τn to put the upper limit of the

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integral as β:

S1 (θ) = S2(θ) ≃[β]∑n=1

(n + 1

2)J0 [(n +

1

2) θ] (6.1)

→β

∫0

λJ0 (λθ) dλ = β2J1 (βθ)βθ

where λ is the continuum extrapolation of n + 12 when we replace the sum by

an integral over all relevant impact parameters.28 Grandy Jr emphasises that

“introduction of a cutoff and direct conversion of the sum to an integral are

gross approximations that can be valid only in the limit β → ∞” (Grandy Jr

2005, pg. 120, my emphasis). The rest of the analysis requires taking fur-

ther asymptotic expansions, such as the Debye asymptotic expansions given

in Grandy Jr (2005, Appendix C). Eventually we obtain the equation for the

rainbow angle:

yR = sinθR2

= (8 + n2c)3n2

cosθ

2= s3

n2(6.2)

It is clear from Grandy Jr’s analysis that we have to be at the limit in order

to make use of these approximations. It is also clear that the mathematics

involved is not surplus structure. There are no obvious mathematical moves

we can make from this equation to recover a wave theoretical model. While

it might seem that we can map from the angle here to the angle in the Mie

solution in some way, I don’t think that such a mapping would be appropriate.

This would have the effect of providing an interpretation of the angle as an

angle between normals to wave fronts, rather than between rays. Given that

the motivations for finding a ray theoretic model from the Mie solution was

to provide a ray theoretic explanation of the colour distribution, adopting a

mapping that results in a wave theoretic interpretation of the angle would not

28With the additional approximation of sin θ by θ in the the Airy diffraction pattern.

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only be unmotivated, but would actually be providing the wrong interpretation

of the angle. Thus we must conclude that the singular limit of β → ∞ does

not result in a model that is surplus structure, and so is a counter example to

the claim that singular limits result in surplus structure.

So how should we make sense of the model we obtain by the singular limit

of β → ∞ on the IC? I think that we have to recognise it as a normal Model

1 obtained via a mapping from an empirical set up. This would require us to

adopt the Mie solution as our empirical set up. Such an empirical set up is

worrying, however. The Mie solution appears to be far too mathematical to be

the starting point for an application of mathematics. While Bueno and Coly-

van claim that the empirical set up need not be entirely free of mathematics,

describing it as “the relevant bits of the empirical world, not a mathematics

free description of it”, and claim that “very often the only description of the set

up available will invoke a great deal of mathematics”, adopting something like

the Mie solution as the empirical set up seems a step too far (Bueno & Colyvan

2011, pg. 354). I can offer a two part response to this worry. First, there is

no need to require the empirical set up to be a completely novel description of

the world. We have already had to apply mathematics to the world in order to

obtain the Mie solution. When we did, we would have mapped the mathemat-

ics of those solutions to a final empirical set up, providing the interpretation

of them. We can adopt this empirical set up as our starting point, and any

complaint that it involves too much mathematics can be answered by pointing

to that previous application (or those previous applications) of mathematics

and how the structures mapped to and from in those respective empirical set

ups were identified. Second, the Mie solution are derived from the Maxwell

equations and are exact solutions. They are as accurate a description of the

world as we can expect from classical mechanical laws. If there is any empir-

ical set up that could be considered the “only description of the set up [that]

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invoke[s] a great deal of mathematics”, then one involving the Mie solution is

an extremely good candidate.

The above considerations prompt an adjustment to when we can claim

surplus structure has been obtained via a singular limit in the context of the

IC. When a singular limit is used to move from Model 1 to Model 2 (in an

iteration of the IC), Model 2 is surplus structure. When a singular limit is

used to move from the initial empirical set up to Model 1 (whether there is

a Model 2 or not), Model 1 is not surplus structure, but a typical Model 1.

This adjustment removes any special features from the singular limits. i.e. A

singular limit no longer entails surplus structure on the IC.

A potential criticism of this solution is that we should not be moving from

the empirical set up to the mathematical domain via a singular limit. This

is motivated by some intuitions: first that singular limits only make sense

between mathematical structures (i.e. they should only take place between

Model 1 and Model 2); and second, and slightly differently, that moving from

the empirical set up to a mathematical model via a singular limit is ‘too quick’.

The idea here is that singular limits require some setting up of the situation

mathematically, before we can perform them. i.e. There are moves we to

take in the derivation before we perform the singular limit of β → ∞. The

first of these intuitions can be rejected by either pointing to the mathematics

involved the empirical set up, or positing that we move to a Model 1 that is

isomorphic to the empirical set up before taking the limit (and do not always

do so explicitly). The second intuition can be assuaged, though not rejected,

by claiming that the empirical set up we use in such a situation comes from

another application of the IC. The reason this is not a conclusive response is

that it is not prima facíe clear whether we could have a suitable application

of the IC where we ‘finish’ with an empirical set up that would satisfy the

requirements of the IC and contain the required structure, i.e. the structure

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of the mathematics at the point in the derivation before we apply the limit.

6.3.2 CAM Approach

The CAM approach is a way of dealing with the slow convergence of the Mie

solution. The approach consists in transforming the Debye expansion of the

Mie solution from an infinite sum of real functions to an integral of complex

functions. The saddle points and Regge poles of these functions are then used

to gain information on how the scattered wave behaves: the saddle points

are related to reflected and refracted light and the Regge poles to surface

waves, waves of light that travel along the surface of the water droplet. Thus

the application of the CAM approach appears to be a standard introduction

of surplus structure at the immersion stage: we introduce the genuinely new

features of the complex plane in order to gain greater insight into the behaviour

of the real functions found in the Mie solution. This occurs due to the shift

from the infinite sum of real functions to the integral over complex functions,

i.e. due to the application of the Poisson sum formula:

∞∑l=0

ψ (Λ, r) =∞∑

m=−∞(−1)m

∞

∫0

ψ (Λ, r) e2imπΛ dΛ (6.3)

The Poisson sum formula is a Fourier transform. These are transformations

that allow us to represent a series (i.e. a sum) of discrete values as a continuum

(i.e. via an integral) and in doing so moves us to the complex plane. Fourier

transforms result in periodic functions, that is they include transformations

to functions that involve sine and cosine functions. The transformations are

due, in part, to the Euler formula, the relationship between the exponential

function and the trigonometric functions:

eiθ = cos θ + i sin θ (6.4)

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This provides a chance to see whether any of the possible solutions might

be capable of solving the dilemma between the Very Large R3 Problem and

the Mathematical Restriction Problem. The situation above looks like it would

be suited to appealing to the ‘natural extension’ solution to the Mathematical

Restriction problem. First, there is the move to the complex plane, which

is the example I used to introduce the notion of natural extension. Second,

recognising that the Poisson sum formula is a Fourier transform and so in part

relies on the Euler formula, suggests that we need the Euler formula in the R3

component of the partial structure of the Debye expansion of the Mie solution.

This does not seem like an unlikely thing to include in the R3 component.

Much as one might argue that the structure of the complex plane should be

included in the R3 for any real function, Euler’s formula should be included in

the R3 for any trigonometric or exponential function; Euler’s formula displays a

relationship that is fundamental to a proper understanding of these functions.

The question remains over what else one would need to include in the R3.

For example, do we need further mathematical techniques to perform Fourier

transforms? A second question is also raised: how much mathematics should

be included in the derivation step? As I will outline below, solving the resulting

equations is not a straight forward matter. Due to the coalescing of the saddle

points, further mathematical techniques (the CFU method) are required to

provide a solution to the equations that can provide suitable physics for the

rainbow. These two questions can be summarised under the question of ‘what

are the family of structures we map into?’ Before answering this question, I

will detail how the surplus structure of the CAM approach is to be related back

to the Mie solution. At best this should involve identifying what interpretation

mapping we should adopt. At minimum it should identify which parts of the

mathematics of the CAM approach should be mapped back to the mathematics

of the Mie solution.

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Interpreting the CAM Approach’s Solutions

The saddle points and Regge poles of the Debye expansion, extended into the

complex plane, are used to obtain the physics of the rainbow. The mathe-

matical details of how we obtain these critical points was set out in the last

chapter, in §5.4, and in the Appendix, in §A.5. The reason behind looking at

the transformations and the targeting of the critical points can be summarised

as follows. The transformation was employed to obtain a series that converged

quicker than the original Mie solution. This required that we adopt the Debye

expansion of the scattering amplitudes, from which we obtain the asymptotic

representation of each term by applying the Poisson sum formula to each term

and then making use of the ability to deform paths in the complex plane. This

results in a background integral, and allows us to “concentrate dominant con-

tributions to the integrals into the neighborhood of a small number of points

in the λ plane, [the] critical points. The main types of critical points are sad-

dle points, that can be real or complex, and complex poles such as the Regge

poles” (Nussenzveig 1992, pg. 62). As stated in §5.5, the interpretation of the

Regge-Debye poles and saddle points varies across the physics text books. The

Regge-Debye poles are consistently referred to as representing surface waves,

but the texts vary as to whether the surface waves are cashed out in terms

of actual waves or as rays taking short cuts along the surface of the droplets.

The saddle points are variously interpreted as points of stationary phase or

as rays. Points of stationary phase are also interpreted as rays (Nussenzveig

1992, pg. 62):

A real saddle point is also a stationary-phase point. Since the phase of

the integrand in most cases can be approximated by the WKB phase,

stationarity implies that the action principle (or Fermat’s principle) is

satisfied, so that such points will be associated with contributions from

classical paths (or geometric-optic rays)

189

As I understand the mathematics, and due to conversations with physi-

cists,29 I think that whether the IC is capable of interpreting the Regge-Debye

poles and saddle points in a way that is consistent with all that I have said

above depends on whether the models are under construction or being used.

This is because there appears to be different concepts required during the de-

velopment of the model compared to its use. Ray theoretic concepts seem

essential to the generation of the uniform CAM approach. Once we have con-

structed the model, some of its uses (explanation, prediction, etc.) can be

performed through interpreting the saddle points as points of stationary phase

and the Regge-Debye poles as actual surface waves. There is at least one

use that requires ray theoretic concepts, however: the process of teaching and

learning the model needs the ray theoretic concepts, or at least relies on them

heavily, as heuristic and pedagogical tools.

If it is the case that ray theoretic concepts are required to interpret the

mathematics of the CAM approach during model construction, this is prob-

lematic for the IC and the partial structures framework. One of the main

draws for adopting the partial structures version of the Semantic View is that

the partial structures framework is supposed to provide a unified way of han-

dling both model construction and model use. Showing that the construction

of the CAM model requires ray theoretic concepts but the use of the repre-

sentation does not require them threatens this unified approach. Bueno and

French outline the partial structures approach to model construction and the

way it relates to the IC as follows (Bueno & French 2012, pg. 88-89):

Using [the partial structures approach], we can provide a framework for

accommodating the application of mathematics to theory construction

in science. The main idea is that mathematics is applied by bringing

29I exchanged a series of emails with John Adams and James Lock, wherein I discussedthe appropriate interpretations of the Regge-Debye poles and saddle points, and the indis-pensability of ray theoretic concepts to these interpretations.

190

structure from a mathematical domain (say, functional analysis) into

a physical, but mathematized, domain (such as quantum mechanics).

What we have, thus, is a structural perspective, which involves the es-

tablishment of relations between structures in different domains. Cru-

cially, we typically have surplus structure at the mathematical level, so

only some structure is brought from mathematics to physics.

. . .

It is straightforward to accommodate this situation using partial struc-

tures. The partial homomorphism represents the situation in which

only some structure is brought from mathematics to physics (via the

R1- and R2-components, which represent our current information about

the relevant domain), although ‘more structure’ could be found at the

mathematical domain (via the R3-component, which is left open). More-

over, given the partiality of information, just part of the mathematical

structures is preserved, namely that part about which we have enough

information to match the empirical domain. These formal details can

then be deployed to underpin the [IC].

However, one can find comments else where in the literature on partial struc-

tures which threatens this straightforward understanding of how model con-

struction should be understood. In their (2012), Bueno and French claim that

the London-London example of superconductivity30 is a case of “model con-

struction” (Bueno & French 2011, pg. 862). In his (2000), French distinguishes

between horizontal and vertical relationships between theoretical models and

structures (French 2000, pg. 106):

[Partial isomorphisms capture] (i) the ‘horizontal’ inter-relationships be-

tween theories, thus providing a convenient framework for understanding

theory change and construction; and also (ii) the ‘vertical’ relationships

30This is a different example to the London account of superfluidity, though very closelyrelated.

191

between theoretical structures and data models, accommodating, in par-

ticular, the role of idealisations

In their (2011) and (2012) Bueno and French indicate that model construc-

tion occurs along the normal IC lines. We construct models by representing

empirical set ups mathematically, perform surrogative inferences in the math-

ematics to obtain new (partial) structures (and partial structural relations) in

the empirical set up, and so have a new model. Neither of these papers refers

to the distinction between the vertical and horizontal relationships between

models that French outlines in his (2000). I think it is ignoring this distinction

which has caused the problem. When discussing the four types of surplus struc-

ture in §6.2.3 I argued that the fourth type, that of accommodating Hesse’s

account of analogies, belonged to the SV rather than the IC. I also argued in

§6.2.1 that the IC does not strictly produce idealised representations of the

world, as the targets of the IC are the empirical set ups, which only become

representational vehicles themselves when used in scientific representations in

the context of the SV. I propose that we understand model creation as part

of the SV, along the horizontal axis (the accommodation of Hesse’s analogies

should also be placed here). One might be tempted to place the IC on the

vertical axis. I think, however, that it should in fact form its own third axis,

due to mathematics being involved in models on both axes. This allows the

IC to account for the mathematical representation that occurs in all parts of

the SV, while not falling prey to problems such as how to relate the empirical

set ups to the world at any one time (as this is done by the vertical axis).

If model construction occurs along the horizontal axis, then we are not

constrained by the strict requirements of the IC. The requirement to iterate

the IC during the CAM approach meant that the saddle points, in Model 2,

could only be related to the wave theoretic Model 1, which itself was mapped

to the empirical set up to gain a physical interpretation. i.e. The mathematics

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in Model 1 was also uninterpreted mathematics (though we can describe this

as wave theoretic as it gains a wave interpretation through the mapping to

the empirical set up). As there was no ray theoretic model in this iteration,

we could not map the saddle points to rays, as we appear to need to do in

order to construct the model. If model creation occurs within the scope of

the SV’s use of partial structures on the horizontal axis, however, then we are

dealing exclusively with models, i.e. interpreted mathematics. In this case,

the requirement to map back through uninterpreted mathematical models is

dropped. This raises the possibility of having more freedom in our choice of

mappings between models and which models we use. For example, we could

map from a model with a wave theoretic interpretation, to a model that has

a ray theoretic interpretation, or from both to a third model which contains

both ray and wave theoretic interpreted components.

The obvious objection to this is that we will result in models that claim

inconsistent things about the world. There are numerous responses available

to inconsistent models, which Bueno and French survey in §5 of their (2011).

They advocate accepting that inconsistent models exist, but reject that they

are false because the inconsistent objects they putatively refer to do not exist.

Rather, they accept the representations to be partially-true. For example, the

object the Bohr model putatively refers to is held to not exist, though the

representation is regarded as partially-true, reflecting its success. The “central

inconsistency” in the model is claimed to be the stable stationary states (Bueno

& French 2011, pg. 867). One can consider two parts of the model that are

inconsistent, the dynamics of the electron in the stationary state and the tran-

sitions between these states, to be considered parts of “strong sub-theories”. At

least in the Bohr case, the inconsistent claim, the stable stationary states, can

be located in the R3 component, to represent statements which have yet to be

established whether they hold in the domain of investigation. If this approach

193

works for handling the use of inconsistent representations, hopefully it can be

adopted to handle the inconsistencies that arise during model creation.

There is a lot more work to pursue here: the notion of the IC being a

third axis requires further exploration, while the ability of the partial struc-

tures framework to accommodate inconsistent models is now a more significant

feature of the framework.

Which Families of Structures?

I suggested above that the only additional mathematics required for the CAM

approach when moving from the Mie solution was the mathematics required to

solve Fourier transformations, i.e. Euler’s equation and mathematics involved

in the extension into the complex plane. The mathematical steps involved

in the application of the CAM approach fall into those conducted within the

wave theoretic model and those conducted within the CAM model. If new

mathematical resources are required than are required to derive the Mie solu-

tion, then further mathematical structure will have to be introduced, and we

might be dealing with more than one case of surplus structure and hence with

more than one iteration of the IC. The most significant steps are: the Debye

expansion, the Poisson sum formula, the deformation of the contour integral to

obtain the Regge-Debye poles and saddle points, and finally the CFU method

to find the saddle point contributions around θ = θR. I will look at each of

these steps and detail whether any additional resources are required.

The Mie solution could be interpreted as involving standing waves within

the water droplet and transmitted plane waves outside of the water droplet.

The Debye expansion involves the transformation of the Mie solution which can

then be interpreted as involving spherically partially transmitted and reflected

waves. This involves expressing the scattering function in terms of spherical

transmission and reflection coefficients. These are expressed in terms of the

194

Henkel and Bessel functions. All of this work can be done within the Mie

model, which is also expressed in terms of Henkel and Bessel functions. No

extra resources are needed. The application of the Poisson sum formula and its

extension into the complex plane constitutes the (partial) mapping from the

wave model to the CAM model, and the introduction of the surplus structure

of the complex plane. The deformation of the contour integral, finding poles

and saddle points are all stand procedures in complex analysis. The resources

needed to conduct these moves should come with the move into the complex

plane.

The CFU method should be regarded as a part of the derivation step. Al-

though there is discussion of introducing “a mapping that preserves the saddle

point structure”, the move is a rather typical one: a change of variables can

hardly be considered to be introducing genuinely new mathematical resources,

and certainly not the introduction of a family of structures. Indeed, the CFU

method only transforms one equation that contains an inexact cubic to one

that contains an exact cubic.

This suggests that although the mathematics involved in the CAM ap-

proach is quite sophisticated, in terms of the families of structures required it

is a rather straight forward approach. We only need to move into the complex

plane and bring the standard tools required for solving and manipulating com-

plex functions: finding poles, constructing residue series and solving integrals

whose contributions come from critical points. This suggests that the Mathe-

matical Restriction Problem is likely to be solved in this instance through the

‘natural extension’ solution, as we need nothing beyond the typical mathemat-

ical structures that come with a move into the complex plane.

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6.4 Conclusion

In this chapter, I have argued that the IC can accommodate the relationship

between faithfulness and usefulness by leveraging the notions of partial truth

and partial structure, adopting an analogous view to the extrinsic and intrin-

sic conception of theories offered by the proponents of the partial structures

version of the SV. I also argued that the structural resources used were the

same for the Galilean and singular limit idealisations, but that the notion of

surplus structure required some work to be accommodated within the IC. I

rejected the idea that singular limits necessarily lead to the adoption of sur-

plus structure. This was shown with the β →∞ idealisation used to obtain a

ray theoretic model of the rainbow. I argued that surplus structure could be

accommodated on the partial structure framework, however, doing so opened

up the account to a dilemma from the Very Large R3 Problem and the Math-

ematical Restriction Problem. I proposed several possible solutions to each

horn of this dilemma, the most promising of which is the ‘natural’ restriction

solution. This solution aims to restrict the R3 through including only the

mathematics in the R3 that could be considered a ‘natural’ extension in the

way that complex numbers could be considered a ‘natural’ extension of the

real numbers. The plausible, general success of this solution was outlined with

its success in restricting the R3 for the CAM approach. The most significant

further work that needs to be carried out is developing the idea of the IC as

being the third axis of the SV approach, where inter-theory relationships are

along the horizontal axis and the relationships between data, phenomena and

theoretical models are along the vertical axis.

I will now investigate what answers the PMA can provide to the questions

concerning the relationship between the faithfulness and usefulness of repre-

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sentations.

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7 | Pincock’s Mapping Account

and the Rainbow

7.1 Introduction

In this chapter I will turn my attention to Pincock’s Mapping Account of

representation, subjecting it to the same questions I posed to the IC: what

is the relationship between the faithfulness of an idealised representation and

the usefulness of such a representation?; and are the structural resources used

by the PMA to accommodate different types of idealisations are the same? I

will argue that a major problem with the PMA is an inability to establish a

general explanation for how usefulness and faithfulness are related. This is in

part due to Pincock’s approach to the content of representations and what he

believes we can infer from the content, and in part due to the position of ‘meta-

physical agnosticism’. This is a position he adopts in response to arguments

concerning emergentist and reductionist interpretations of singular perturba-

tion representations. The ‘metaphysical agnosticism’ position raises questions

over the generality of Pincock’s account and whether Pincock’s methodology

is compatible with Contessa’s approach towards faithful epistemic representa-

tion.

I will again make use of the two idealisation examples from the rainbow,

the β → ∞ singular limit and the move to the complex plane involved in the

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CAM approach. Pincock discusses this example himself. He makes compar-

isons between the β → ∞ idealisation, the k → ∞ and his attitude towards

singular perturbations. I will argue that this comparison is not informative

with respect to metaphysical agnosticism. Pincock also contrasts his views

on the CAM approach to Belot’s and Batterman’s position on the rainbow.

I explore Pincock’s metaphysical attitude towards the CAM approach and

conclude that he does not hold a metaphysical agnosticism position towards

it. Rather, I suggest that he has adopted a kind of reductionist position to-

wards the critical points involved in the CAM approach. Pincock’s discussion

of Belot’s and Batterman’s positions is also unhelpful for understanding his

metaphysical agnosticism position. I conclude the chapter by arguing that

metaphysical agnosticism is either too undeveloped to hold as a serious posi-

tion or collapses into a form of reductionism.

7.2 Pincock’s Mapping Account and

Idealisation

Idealisation is, in general, dealt with very straightforwardly on Pincock’s ac-

count by his various notions of content, the flexibility in what structural re-

lations he admits, and the specification relation. Despite this ease of dealing

with idealisations in general, I will argue that Pincock’s approach threatens a

unified account of how faithfulness and usefulness are related on the PMA.

7.2.1 Account of Idealisation

The majority of idealisations will be accounted for by enriched contents, struc-

tural relations that involve mathematical terms and the specification relation.

How these are handled is best explained by Pincock himself, when he intro-

duces the enriched contents (Pincock 2012, pg. 31):

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In the heat equation, we have to work with small regions in addition

to the points (x, t) picked out by our function. We should take these

regions in the (x, t) plane to represent genuine features of the temper-

ature changes in the iron bar. This representational option is open to

use even if the derivation and solution of the heat equation seem to

make reference to real-valued quantities and positions. We simply add

to our representation that we intend it to capture temperature changes

at a more coarse-grained level using regions of a certain size that are

centered on the points picked out by our function. The threshold can

be set using a variety of factors. These include our prior theory of

temperature, the steps in the derivation of the heat equation itself, or

our contextually determined purposes in adopting this representation to

represent this particular iron bar. [Pincock’s] approach is to incorporate

all of these various inputs into the specification of the enriched content.

The enriched content has a much better chance of being accurate as it

will typically be specified in terms of the aspects of the mathematical

structure, which can be more realistically interpreted in terms of genuine

features of the target system.

. . .

We arrive at the enriched content of a representation, then, by allowing

the content to be specified in terms of a structural relation with fea-

tures of the mathematical structure beyond the entities in the domain

of the mathematical structure. The resulting structural relations need

to be more complicated than just simple isomorphisms and homomor-

phisms. In particular, we allow the specification of the structural relation

to include mathematical terminology. For example, we may posit an iso-

morphism between the temperatures at times and u in the mathematical

structure subject to a spatial error term ε = 1mm.

More sophisticated idealisations will involve schematic contents. The ex-

ample of the deep water waves shows that not all cases of taking singular limits

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involves a case of singular limit idealisation.1 Remember that a singular limit

idealisation has to necessarily involve the limit. The deep water wave case is

justified along Galilean lines, and as such the limit can be removed. Yet Pin-

cock argues that the deep water wave idealisation should not be understood as

the taking of a limit at all, but rather in terms of taking a set of scales. I shall

call this sort of idealisation perturbation idealisation. This itself can be split

into two types: regular and singular perturbation idealisation. Pincock argues

that at least some cases of singular limit idealisation should be understood in

terms of singular perturbation idealisation. In doing so, one can solve some of

the problems introduced by Batterman, namely those of interpreting the limit.

I will introduce the deep water wave example, first focusing on how it is justi-

fied in terms of a Galilean idealisation, then on how Pincock argues for it to be

understood as a regular perturbation idealisation. I will then summarise how

the notions of content and the specification relation are involved in a singular

perturbation idealisation. This will allow me to move onto a discussion of how

faithfulness and usefulness are related on Pincock’s account.

The deep water wave representation aims to find the speed of a wave crest.

This involves starting with the Navier-Stokes equation, applying the ‘small

amplitude’ idealisation, then the deep water wave idealisation to find a simple

representation in terms of the wavelength of the wave. When introduced as

a Galilean idealisation, the deep water wave idealisation involves taking the

variable for the depth of the ocean, H, to infinity. We take H to infinity

because we obtain a tanhx term, where x = 2πHλ , and as x → ∞, tanhx → 1.

Thus, when the depth of the ocean is much greater than the wavelength of the

waves, such as when H > 0.28λ, we can replace the tanhx term with 1 and

obtain a maximum error of 3%. Remember that on the PMA, the mathematics

has an interpretation. This would seem to imply that taking H →∞ involves

1See Pincock (2012) pg. 96-104 for the full details of Pincock’s analysis of this example.

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the claim that the ocean is now infinitely deep. This is not the case, however

as taking the limit involves decoupling the interpretation from H, leaving us

with an unspecified parameter. Thus the representation can be described as

having nothing to say about the depth of the ocean after taking the limit.2

Decoupling interpretations also involves the specification relation changing (as

this supplies the interpretation). Justifying an idealisation through appealing

to a maximum error also implies the adoption of a structural relation which

includes such an error term. Thus the only difference between the simple cases

of idealisation mentioned above that involved the enriched content is the move

to schematic content and the lack of interpretation for the mathematics that

constitutes the schematic content.

The main motivation Pincock offers for understanding such an idealisation

(and singular limit idealisations) in terms of perturbation idealisations is that

we can replace the “vague notion of an approximately true claim with the claim

that for phenomena p some set of scales s is adequate” (Pincock 2012, pg. 103).

This is promising for getting a grip on the relationship between faithfulness and

usefulness, given that the IC cashes out approximate truth in terms of partial

truth. As I argued above, partial truth and the associated partial structures

are capable of providing an explanation of how faithfulness and usefulness

are related. To understand the deep water wave example in terms of a regular

perturbation, Pincock first introduces us to the notion of scales. Here a variable

is replaced by a ‘scaled’ version of it. For example, the x∗ variable (* denoting

an original variable) is replaced by x = x∗

λ . There is often a choice in the set

of scales, that is, which variables we choose to replace and which variable we

divide the original by. Our choice of x and λ constitutes the claim that the

“relevant processes operate in the x spatial direction only on the order of λ”.

The idea is that we are attempting to show that the contribution of certain

2The other interpretation of schematic content is that these are pieces of the mathematicswhich have no interpretation. See §4.2.2 - Pincock’s Notion of Content.

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terms are negligible relative to other terms (preferably that non-linear terms

are negligible relative to linear terms, for ease of calculation). Notice that

our chosen replacement also results in x becoming dimensionless (it involves a

length variable divided by a length variable). The set of scales chosen for the

deep water wave example result in the claim that (Pincock 2012, pg. 103):

terms preceded by h become orders of magnitude more important, or

equivalently, that terms preceded by 1h =

λH become orders of magnitude

less important. When h appears in the boundary conditions, this has

the effect of moving the boundary as far from the system as possible.

Thus for this set of scales to be adequate, boundary effects must be

irrelevant to what we aim to represent

We establish whether a set of scales is adequate by seeing whether the set of

scales “reveals anything of relevance to p”, and then conducting “additional

tests . . . to see if the representation does indeed accurately capture” p. If it

does, then we have an adequate set of scales.

In summarising his discussion of why sets of scales (and by implication,

perturbations) are important, Pincock gives the clearest description of how he

thinks faithfulness and usefulness are related (Pincock 2012, pg. 104):

What we see, then, is a kind of trade-off between the completeness of a

representation and its ability to accurately represent some phenomena of

interest. . . . The problem is basically that a complete representation of a

complex system will include countless details, and these details typically

obscure what we have selected as the important features of the system.

By giving up completeness, and opting for a set of scales, we shift to a

partial representation of the system that aims to capture features that

are manifest in that scale. This partiality gains us a perspicuous and

accurate representation as well as an understanding of features that

would otherwise elude us.

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The idea here seems to be that by adopting a particular set of scales we are

introducing some mathematical structure that partially represents the target

system and in doing so reveals certain features at that scale. Thus we will be

relating part of the target system to our mathematical structure. This will

involve the structural relation mapping from the mathematics to only part

of the target system, with the specification relation helping us to interpret

the mathematical structure appropriately for the scale we are using. i.e. That

parts of the mathematics should not be interpreted at this scale (are schematic

contents) in the way they would in a mathematical structure that is a complete

representation.

In order to make the notion of “adequate set of scales” more precise, Pin-

cock turns to perturbation theory.3 Our original problem involved finding an

unknown function f(x) that satisfies some differential equations and boundary

conditions. We change it by recasting f(x) as a function of f(x, ε), where ε

is a value that is small for the domain in question. We then aim to find an

asymptotic expansion of f(x, ε) to N terms, and use the magnitude of terms

over a certain order of the asymptotic expansion of the function to establish

which terms of the asymptotic expansion are to be kept and which are to be

rejected. This allows us to find an answer to any degree of accuracy. i.e. the ε2

term might be smaller than experimental accuracy, so all ε terms of the order

greater than 2 can be rejected.

A regular perturbation is one where we can start with ε = 0, and add

correction terms from the asymptotic expansion (i.e. where the order of ε >

0). A singular perturbation, however, occurs when there is a difference in

qualitative character between the ε = 0 and ε ≠ 0 cases. Pincock uses the

example of the equation εm2 + 2m+ 1 = 0. When ε = 0, this equation has roots

m = −12 , but when ε ≠ 0, the equation has two roots which cannot be recovered

3The details can be found in his (2012), pg. 104-105.

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from the ε = 0 case (they are in fact undefined when ε = 0). In such a case,

we shift to singular perturbation theory and have to employ multiple scales.

What is nice about Pincock’s moves here is that he has given us a formal

framework within which to understand the difference between the limits in

Galilean idealisations, which can be removed, and those which produce singular

limits, such as in Batterman’s examples. If we can use regular perturbations,

then we have a Galilean limit, whereas if we have to singular perturbations,

we have singular idealisation type limits.

Singular perturbations result in a different story of how the mathemati-

cal structure, specification relation and structural relations meet. Whereas in

the regular perturbation case, one merely introduces some schematic content

(and hence that part of the mathematics is decoupled from its interpretation),

singular limits result in the need to give the resulting representations “a differ-

ent physical interpretation than the original representations . . . the singular

character of the limit can be linked to the need to offer an interpretation in

terms of different physical concepts” (Pincock 2012, pg. 223). I take this to

mean that a singular limit causes all of the content to become (momentarily)

schematic, before being given a new interpretation (i.e. a new specification re-

lation), though some schematic content will obviously stay schematic (such as

the term(s) involved in the singular limit/parameter that produces the singular

perturbations).

7.2.2 PMA, Faithfulness and Usefulness

The above discussion has made it clear that Galilean idealisation requires dif-

ferent structural moves to singular (perturbation) idealisation on Pincock’s

Mapping Account. This is the answer to the second question posed at the start

of this chapter.4 Due to these differences in structural moves, there might be

4Are the structural resources used by the PMA to accommodate different types of ide-alisations the same?

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differences in how faithfulness and usefulness are related in each case. I will

look first at Galilean idealisation, and then at what Pincock says about inter-

preting three examples of singular perturbations: the boundary layer theory, a

damped harmonic oscillator, and Bénard cells. Pincock advocates adopting a

position of metaphysical agnosticism towards the damped harmonic oscillator

and Bénard cells. I will explore what this position entails for the relationship

between faithfulness and usefulness on the PMA.

The matter is straightforward for Galilean idealisation. Such an idealisation

is partially faithful due to the structural similarity between the enriched (and

genuine) contents and the target, cashed out by the (mathematically infused)

structural relation. The usefulness is then accounted for by the specification

relation and how this alters the structural relation. Remember that Pincock

wants to include such factors as what we intend our representation to capture

(i.e. temperature at a coarse-grained level), contextual determined purposes

of adopting a particular representation, and so on, into the specification of

the enriched content, i.e. the specification relation (Pincock 2012, pg. 30-

31).5 Being reductive, we can describe Galilean idealisations as being useful

on the PMA because we design them to be: either we adopt the appropriate

contents for our theoretical concepts we are trying to represent (e.g. enriched

contents and temperature), or due to the mathematics we wish to use, and

adopt a suitable structural relation for these purposes (a structural relation

that contains something similar to an error term that allows for more useful

representations to be given). The general idea is to obtain a structure that

highlights relevant features.

The relationship between faithfulness and usefulness is much more com-

plicated in the case of singular perturbation idealisations.6 The most obvious

5See my discussion of this in §4.2.2 - Pincock’s Notion of Content and of the specificationrelation in §4.2.2 - Pincock’s Structural Relations and Specification Relation.

6While Pincock claims that the “vague” notion of approximate truth can be made moreprecise by the notion of an adequate set of scales, I reject this as being useful for the notion

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complicating factor is that these representations are likely to have a high quan-

tity of schematic content. While this might seem like it would make things

easier (as there would be less interpreted content that could be described as

faithful), this is not the case as we also have the recoupling of some of this

content in terms of different physical concepts than those involved in the initial

representation.

Boundary Layer Theory

One of the key techniques involved in fluid dynamics is the boundary layer

theory.7 Here, the flow of a fluid around an object is analysed. One wishes

to find out how the pressure and velocity values for the fluid are arranged

around the object. To do so one can either adopt the Navier-Stokes equations

(though these turn out to be intractable for this problem) or the Euler equa-

tions and adopt a set of scales. Unfortunately, the second option results in the

prediction of zero drag on the object, which can be shown to be false through

simple experiment (Pincock 2012, pg. 108-109). The cause of this problem is

the assumption that the set of scales is adequate for the whole domain. By

splitting the domain into two and adopting a sets of scales for each region

of the domain, we can find a solution by matching the sets of scales at the

boundaries of the regions. The region which is closest to the object is called

the boundary layer, and its set of scales is picked using a new quantity δ,

which is described as the width of the boundary layer (Pincock 2012, pg. 110).

This representation is very successful. Yet, although “the edge between the

boundary layer and the outer region is fundamental to the representation, . . .

we do not take it to represent any genuine edge in the system itself”, and there

are in fact several inconsistent ways of establishing the value of δ (Pincock

of faithfulness. The way in which Pincock cashes out the notion of adequacy is binary, ratherthan gradable.

7See §5.6 of Pincock’s (2012) for his presentation and discussion of the boundary layertheory. He relies on Kundu & Cohen (2008), §10 for his presentation.

207

2012, pg. 112), (Kundu & Cohen 2008, pg. 346-348). This prompts Pincock

to claim that there is no warrant for a metaphysical interpretation of the edge

between regions (i.e. that there are two regions in any physical sense). He

also promises that he will go on to argue that “we have no general reason to

conclude that the accuracy of [singular perturbation idealisations reveal] new

underlying metaphysical structure” (Pincock 2012, pg. 113). Thus the lesson

we should take from the boundary layer example is that singular perturbations

can produce mathematics of purely instrumental benefit: they allow us to ob-

tain an accurate result but with no new physical structure being uncovered.

This would suggest that the link between faithfulness and success is broken

here, in that the content of the representation is wholly schematic.

One might wonder how we are to get accurate results out of a representa-

tion which contains wholly schematic content. The answer to this is to recall

Pincock’s distinction between intrinsic and extrinsic mathematics, core con-

cepts, and that we move from schematic content to genuine content in the

process of producing a prediction. In order to obtain a prediction from a rep-

resentation with schematic content, we must specify parameters according to

the target system and in doing so we move from schematic content to genuine

content (Pincock 2012, pg. 32). The solutions to equations are not always part

of the content of our representations (Pincock 2012, pg. 38). In this instance

I must conclude that there is no link between the faithfulness of a represen-

tation and its usefulness. It is tempting to call such representations entirely

unfaithful: if the content of the representation is wholly schematic, then it

can be understood as saying nothing about the target system. In such an in-

stance, the mathematics could be argued to not even constitute an epistemic

representation as it would no longer be interpreted in terms of the target. The

mathematics would still be taken to be denoting the target. The adoption of

genuine content might solve this problem, providing an interpretation of the

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mathematics in terms of the target again. However, there does not appear to

be any link between the representation with genuine content and the target

that could satisfy the requirements of answering how the partially faithfulness

of the representation and its usefulness is related. The representation is useful

due to the schematic content being made into genuine content through the

filling in of specific parameters.8

The next two examples, the damped harmonic oscillator and the Bénard

cells, are two examples of a different type of singular perturbation. The bound-

ary layer theory involves splitting the domain into two parts, with a set of scales

for each region. In the damped harmonic oscillator and Bénard cells exam-

ples the domain remains the same, but a different set of scales is adopted for

different time periods.

Damped Harmonic Oscillators

For the damped harmonic oscillator, we start with the following equation:

my′′ + cy′ + ky = 0 (7.1)

If we assume that the damping effects are small compared to the dominant

process, we might adopt a set of scales such as:

t = t∗√mk

(7.2)

However, as time increases, this set of scales fails (Pincock 2012, pg. 115). Thus

we should conclude that “we cannot assume there is a single process operating

on a single time scale”, and think of the y as a function of two variables tF = t

and tS = εt. When ε << 1, tS << tF so tS “corresponds to a ‘slow’ time scale”.

The damped harmonic oscillator example is used by McGivern (2008), who8The worries here not unique to Pincock’s account: similar concerns will arise on any

account for any instrumental representation.

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argues in favour of a physical notion of emergence against ‘micro-based physi-

calism’.9 In general, McGivern argues that the multi-scale analysis present in

the damped harmonic oscillator involves properties that do not fit into Kim’s10

conception of reduction (McGivern 2008, pg. 54). Kim’s conception involves

identifying all higher-level properties with distinct micro-based properties,

understood as mereological configurations of lower-level micro-constituents.11

McGivern’s position can be summarised as the claim that multiscale analysis

“often involves decomposing a system’s behavior into components operating

on different scales” as opposed to “explaining a system’s behavior by relating

it to the behavior of its micro-constituents” i.e. physical, mereological decom-

position.

The multi-scale structure outlined above in the damped harmonic oscillator

is a high-level structure. McGivern sets out the approach a micro-physical

reductionist should take towards this property (McGivern 2008, pg. 64):

To accommodate the kind of multi-scale structure found in multi-scale

analysis within the standard framework of ‘micro-based’ levels, we would

need to show how multi-scale structural properties can be identified with

distinct micro-based properties and then cashed out in terms of ‘specific

9McGivern does not actually argue that reductionism is false, but that arguments for itmust be given “in terms specific to the properties, explanations, and theories involved, ratherthan in the broad terms characteristic of arguments about causal competition” (McGivern2008, pg. 55). One might read the position McGivern adopts here as similar to the onePincock advocates, that we “should use multiscale representations to help reform and sharpenthe metaphysical positions, rather than using them to champion one or the other side”. Iwill look at this briefly below, when considering if Pincock adopts a “wait and see” agnosticposition.

10See Kim (1998, 2003).11As McGivern explains it, Kim’s account starts by characterising the property to be

reduced “in terms of its ‘functional’ or causal role and then identifying that property withwhatever property fills that role on a given occasion. Importantly, these properties areassumed in general to be ‘micro-based’ properties, where a micro-based property is theproperty of having particular parts which themselves have particular properties and standin certain relations” (McGivern 2008, pg. 58). For example, ‘being a water molecule’ is ‘theproperty of having two hydrogen atoms and one oxygen atom in such-and-such bondingrelationship’. Causation need not only occur at the micro-level however. ‘Being a watermolecule’ is a higher-level property of a higher-level entity, i.e. a molecule. Thus we canhave causation at higher-levels, between higher-level entities. Kim also carries out thisreduction for structural properties (Kim 1998, pg. 117-118).

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mereological configurations’ of micro-level entities and their properties.

If one were to explain the structure of an organism in this way, one would

identify the structure of the organism with its cellular structure, then this

structure with its component parts and so on until one reaches the genuinely

micro-based properties. And now McGivern comes to the crux of the issue

(McGivern 2008, pg. 64-65, my emphasis):

In “structural” terms, we can informally characterize this as “the property

of having parts S and F ” where S is the slow scale component and F is

the fast scale component. However, unlike in more familiar cases, such

as those involving cellular structure, the decomposition into fast and

slow components doesn’t give us spatial parts-instead, the decomposition

is of a different “non-spatial” sort. Hence it’s difficult to see how these

could be properties of “non-overlapping” parts, as in the case of micro-

based properties. . . . But in the case of decomposition into fast and slow

components, it is the same particles at the micro-level that realize both

the fast and the slow components: there is no division of the system into

“fast” and “slow” particles. Hence we can’t identify the slow component

with one configuration of micro-level entities and the fast component

with another, even if “configuration” is conceived of dynamically to allow

for the fact that we are dealing the behavior of a system over time. So

we appear to have a kind of structural property that doesn’t fit with the

standard micro-based account of reductive levels, understood in terms

of entities related by (“spatial”) parthood and characterized exclusively

by micro-based properties.

And again later in the paper (McGivern 2008, pg. 70):

But multi-scale structural properties aren’t [spatial]: they don’t involve

a move from the relatively large to the relatively small. Hence, the kind

of thinking that usually motivates the belief that macro-level properties

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can be identified with micro-based ones - the repeated explanation of

phenomena in terms of smaller and smaller entities - doesn’t get off the

ground in the case of multi-scale structure.

Thus we can see how one might understand multiscale representations as not

being capable of being reduced to micro-level properties, at least according to

Kim’s account. McGivern’s arguments are not a complete rejection of reduc-

tion however. He emphasises that he hasn’t tried to argue against reductionism

in philosophy of mind (McGivern 2008, pg. 73). Rather his point is that “it

seems wrong to begin with an assumed structure of ‘levels’ and then try to

fit our theories and explanations to that structure: instead, we need to begin

with our theories and explanations and see what sort of structure they imply”,

that is, we should start with our physical theories and attempt to read our

metaphysical views on structures and levels from those theories (McGivern

2008, pg. 74).

Bénard Cells

Bénard cells are used as an example by Bishop (2008) to argue in favour of

“downward causation”, against Kim’s physical reductionism. Bénard cells are

a stable configuration of a heated liquid. A liquid is sandwiched between two

plates. The bottom plate is heated while the top plate is kept at a constant

temperature. If the temperature gradient between the top and bottom plates

is large enough, then the relevant dynamics of the fluid system shifts from the

diffusion process, which occurs over a short time scale, to the faster convective

process (Bishop 2008, pg. 236) (Pincock 2012, pg. 116). The diffusion pro-

cess results in small scale effects, while the convective process results in large

scale effects. The stability of the diffusion process is broken during the heat-

ing process, though stability is regained with the Bénard cells. The cells are

“large-scale features of the fluid system that influence how the small-scale fluid

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elements move”. The idea here is that we need two scales, one for the diffusion

process and one for the Bénard cells. The size of the parameter we choose in

establishing these scales informs us as to which set of scales dominates. The

Reynolds number is adopted as the parameter; when it is small the diffusion

process dominates, when it is large the cells dominate.

Bishop adopts a view of emergence that includes a notion of downward cau-

sation put forward by Thompson and Varela (Bishop 2008, pg. 230), quoting

(Thompson & Varela 2001, pg. 420):

(TV) A network, N , of interrelated components exhibits an emergent

process, E, with emergent properties, P , if and only if:

(a) E is a global process that instantiates P and arises from the non-

linear dynamics, D, of the local interactions of N ’s components

(b) E and P have global-to-local (‘downward’) determinative influence

on the dynamics D of the components of N

And (possibly):

(c) E and P are not exhaustively determined by the intrinsic proper-

ties of the components ofN , that is, they exhibit ‘relational holism’

This proposal includes the claim that a property emerges not just qua property,

but is instantiated in a process or some other “dynamical ‘entity’ unfolding in

time”.

Bishop goes on to argue that Bénard cells display features indicative of

emergent behaviour. More generally, he argues that nonlinear systems require

some kind of “particular global or nonlocal description”, as the “individual

constituents cannot be fully characterized without reference to larger-scale

structures of the system”, due to the principle of linear composition failing

(Bishop 2008, pg. 231). In particular, he claims that Bénard cells exhibit the

behaviours of control hierarchies and constraints (Bishop 2008, pg. 237):

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The Rayleigh-Bénard system clearly exhibits the features listed in Sect.

2.2 [(Bishop 2008, pg. 232)]. As ∆T exceeds ∆Tc, the homogeneity of

the distribution of the fluid elements and some of the spatial symme-

tries of the container are broken and the fluid elements self-organize

into distinguishable Bénard cells. There is a hierarchy distinguished

by dynamical time scales (molecules, fluid elements, Bénard cells) with

complex interactions taking place among the different levels. Fluid el-

ements are situated in that they participate in particular Bénard cells

within the confines of the container walls. The system as a whole dis-

plays integrity as the constituents of various hierarchic levels exhibit

highly coordinated, cohesive behavior. Additionally, the organizational

unity of the system is stable to small perturbations in temperature and

adapts to larger changes within a particular range.

Furthermore, Bénard cells act as a control hierarchy, constraining the

motion of fluid elements. Bénard cells emerge out of the motion of fluid

elements as ∆T exceeds ∆Tc, but these large-scale structures determine

modifications of the configurational degrees of freedom of fluid elements

such that some motions possible in the equilibrium state are no longer

available.

Bishop argues that the fluid elements are necessary but insufficient to ac-

count for the existence and dynamics of the Bénard cells, or even their own

motions. The local dynamics are constrained by the large-scale structures.

Together the fluid elements, Bénard cells and the system wide forces form the

necessary and sufficient conditions for the behaviour of the system (Bishop

2008, pg. 239). Thus he claims thats that Bénard cells satisfy (a), (b) and (c)

of the TV proposal of downward causation (Bishop 2008, pg. 240):

Bénard cells arise from the dynamics of fluid elements in the Rayleigh-

Bénard system as ∆T exceeds ∆Tc, where each fluid element becomes

214

coupled with every other fluid element (element (a) of TV). More impor-

tantly, Bénard cells act as a control hierarchy, constraining and modify-

ing the trajectories of fluid elements; that is, Bénard cells have a “deter-

minative influence” on the dynamics of the fluid elements as lower-level

system components (element (b) of TV). Moreover, although the fluid

elements are necessary for the existence of Bénard cells, the former are

insufficient to totally determine the behavior of the latter. This relation-

ship between necessary and sufficient conditions seems implicit in (b),

but due to its importance in issues surrounding reduction and emer-

gence, this relationship should be brought out explicitly (e.g., Bishop

(2006); Bishop & Atmanspacher (2006)).

This relationship is a nonquantum kind of “relational holism””understood

in Teller’s (1986) sense, where, the relations among constituents are not

determined solely by the constituents’ intrinsic properties. The proper-

ties of integrity, integration and stability exhibited by Bénard cells are

relationally dynamic properties involving the nonlocal relation of all fluid

elements to each other (element (c) of TV). . . . However, the behavior

of Bénard cells as units differs from holistic entanglement in quantum

mechanics in the sense that fluid elements may be distinguished from

each other while they are simultaneously identified as members of par-

ticular Bénard cells and participate in interaction with fluid elements

throughout the system.

The key difference between the boundary layer example and the damped har-

monic oscillator and Bénard cells examples is that we are not spatially dividing

the domain, but claiming that there are two processes operating at different

time scales.

215

Metaphysical Agnosticism

Bishop and McGivern appear to have presented good arguments which indi-

cate that we should reject reductionism (at least the sort offered by Kim’s

account) for systems that can be represented through temporal multiscale rep-

resentations. Bishop takes a rejection of reductionism for the Bénard cells to

justify a rejection of reductionism completely. McGivern thinks that we are

only justified in rejecting reductionism for temporal multiscale representations

and is open to adopting reductionism in other systems. One’s response to these

arguments will depend on whether one considers there be other options avail-

able. Pincock argues that we do have another option, namely his ‘metaphysical

agnosticism’ (Pincock 2012, pg. 119).

Pincock claims that his metaphysical agnosticism is a third way between

adopting a “metaphysical interpretation”, i.e. reductionism or emergentism,

and what he calls “instrumentalism”. That is, he offers a third table entirely

with reductionism and emergentism the options on the metaphysical table,

instrumentalism a second table and his “metaphysical agnosticism” the third

table. Metaphysical agnosticism involves (Pincock 2012, pg. 119-120 (my em-

phasis throughout)):

emphasiz[ing] the epistemic benefits that multiscale representation af-

fords the scientist. On this picture, a successful multiscale representation

depends on genuine features of the system. We come to know about these

features when a multiscale representation yields a successful experimental

prediction. In this respect, it goes beyond the instrumentalist position.

But it does not end up with a reductionist or emergentist metaphysical

interpretation because the epistemic approach is consistent with a re-

jection of both reductivism and emergentism. To see why, consider the

Bénard cells. . . . [The] epistemic point about what we know based on

our limited access to the details of the fluid dynamics need not entail

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any metaphysical conclusions about the existence of the Bénard cells in

some more robust sense. Clearly, we can only understand the system

by appeal to these larger-scale structures. But this may be a product

of our ignorance and in the end may not correspond to what a fuller

understanding of the system would reveal. . . . More generally [Pincock]

suggest[s] that we cannot move from the success of a multiscale represen-

tation to the conclusion that it reveal novel metaphysical features of the

physical system. This metaphysical agnosticism is consistent with delin-

eating a crucial epistemic role for the multiscale techniques in producing

scientific knowledge.

I take Pincock to be arguing that we can conclude some novel physical fea-

tures of a system from successful multiscale representations (first emphasis),

but that we cannot (in general) use such representations to make novel meta-

physical claims (second emphasis). Such a position would require some of the

content of the representation to be given a new physical interpretation. This

would be a limited interpretation as we would be unable to infer anything

metaphysically novel about the target system from this interpretation, e.g.

whether this feature is due to emergence. Thus we have a case where singular

perturbation theory results in a representation that is not purely schematic

content, as both of our sets of scales have physical interpretations, though

they are limited in the sense just described. This would involve a limited

specification relation and an appropriately restricted structural relation.

There are some major differences between the boundary layer theory, and

the damped harmonic oscillator and Bénard cells, the most important of which

is whether the multiple sets of scales represent genuine features of the target

system. In the boundary layer theory, the reasons for doubting the reality

of the layer are that there are inconsistent methods for setting its width and

(Pincock implies) that the scales are due to a physical, i.e. spatial, split of

the domain. Whereas in the other two examples, the two sets of scales are

217

thought to represent two genuine processes due to the accurate predictions

they give (and possibly as they involve a difference in time, rather than spatial

scale). As quoted above, Pincock’s metaphysical agnosticism commits us to a

genuine feature of the target system “when a multiscale representation yields a

successful experimental prediction”. The boundary layer theory, however, gave

very good predictions, yet Pincock states that “we have no general reason to

conclude that the accuracy of these [multiscale] representations reveals any new

underlying metaphysical structure” (Pincock 2012, pg. 113). If we accept that

one can maintain a distinction between a genuine feature of the system and an

“underlying metaphysical structure”, then I think that Pincock’s metaphysical

agnosticism involves an explanatory gap that threatens his account’s ability to

explain how faithfulness and usefulness are related. Accepting for the sake of

argument that we can admit a genuine feature of the system while having its

metaphysical qua reductionist or emergentist origins being unknown, there is

still the question of how the structure of this genuine feature identified by the

representation is related to the structure of the target system (the metaphysical

origin being one way of establishing this relationship). Establishing how the

structure of the genuine feature identified by the representation is related to

the target system will provide the explanation of how the representation can

be as partially faithful as it is, and yet still be useful.

The problem here is that I do not think Pincock’s arguments against either

what he calls the “instrumentalism” view of the mathematics or against the

metaphysical line are at all convincing. Further, I do not see how, prima

facíe, his position is clearly distinguished from Batterman’s. I will now outline

the problems I have with Pincock’s arguments against the instrumentalist and

metaphysical positions. I will discuss the relationship between Pincock’s and

Batterman’s positions in §7.3.2 - Pincock, Batterman & Belot.

Pincock describes instrumentalism about the mathematics as the following

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attitude (Pincock 2012, pg. 119):

[R]escaling with multiple scales is just a tool that allows us to handle

otherwise intractable mathematical equations. We have learned how to

work with this tool, but it need not reveal anything about the underly-

ing features of the systems represented. Multiscale modelling, then, is

just a piece of brute force mathematics whose larger significance can be

ignored.

This view is similar to one way in which surplus structure can be used: we

adopt some further mathematics to shed light on some intractable problem

from our initial representation of the target system. However it is not entirely

the same, as in some cases the use of surplus structure can indirectly reveal

underlying features of the systems being represented, such as group theory

highlighting the role of symmetry in quantum mechanics.12 The instrumen-

talist position justifies the use of the mathematics pragmatically, by providing

a way to solve otherwise intractable mathematics. There is no question over

the faithfulness of the mathematics, as it is not representing anything on the

instrumentalist view. The surplus structure view has already been accounted

for, during the discussion of the IC.

Pincock rejects the instrumentalist position by claiming that “successful

multiscale representation depends on genuine features of the system”, and that

“we come to know about these features when a multiscale representation yields

a successful experimental prediction”. His rejection of instrumentalism is tied

to his commitment to his metaphysical agnosticism, as he rejects the dichotomy

between the instrumentalist and metaphysical (reductionism and emergentism)

positions. But merely pointing to accurate prediction is insufficient here, as

instrumental uses of mathematics can produce accurate predictions. In order12I argue below in §7.3.2 - Pincock, Batterman & Belot that Pincock has conflated the

instrumentalist and surplus structure positions, given what he says about Belot’s and Red-head’s response to the rainbow.

219

to reject the instrumentalist position and the dichotomy between it and the

metaphysical positions, there must be more work done to establish how multi-

scale representations can depend upon genuine features in a way which these

two broad positions cannot account for.

Perhaps we are to consider the metaphysical agnosticism as a ‘wait and

see’ type of agnosticism. This understanding of the agnosticism has some

textual support.13 But this would not constitute a real rejection of both the

instrumentalist and metaphysical positions. McGivern’s position is prima facíe

the same as Pincock’s. McGivern claims that arguments for reduction should

properly be given in terms that are “specific to the properties, explanations, and

theories involved, rather than in the broad terms characteristic of arguments

about causal competition” (McGivern 2008, pg. 55). He also claims that “it

seems wrong to begin with an assumed structure of ‘levels’ and then try to

fit our theories and explanations to that structure: instead, we need to begin

with our theories and explanations and see what sort of structure they imply”

(McGivern 2008, pg. 74, my emphasis). He doesn’t reject reductionism in toto,

just a version of reductionism that requires spatial decomposition for systems

that require (time) multiscale representation. One can therefore understand

McGivern as being open to the possibility of reductionism in other systems,

and Pincock recognises this (Pincock 2012, pg. 117). McGivern’s position

towards reductionism therefore is a “wait and see” position, asking for further

developments in the reductionist position to occur. If they do not occur, or are

shown to be incapable of occurring, then we are entitled to rule it out as a viable

option. Pincock rejects McGivern’s position, however, taking McGivern to

draw metaphysical conclusions when he endorses emergentism, apparently due

13Pincock talks of the understanding of the system being dependent upon the large-scalefeatures (of the Bénard cells) being a possible result of our ignorance “and in the end maynot correspond to what a fuller understanding of the system would reveal”. He also claimsthat “[e]ither metaphysical position seems premature, even if we can reform these views toaccommodate this kind of case” (Pincock 2012, pg. 119).

220

to understanding McGivern as arguing for the existence of emergent properties

due to the genuine causal and explanatory power they are attributed14 (Pincock

2012, pg. 117-119). The distinction between Pincock and McGivern, then, is

that Pincock rejects reductionism and emergentism in toto, whereas McGivern

thinks that they are still viable positions. Thus Pincock’s agnosticism cannot

include any sort of “wait and see”, contra McGivern’s position. Further, any

inclusion of a “wait and see” attitude in his agnosticism would contradict his

more general point that we cannot move from the success of these multiscale

representations to novel metaphysical features.15

Given the above arguments, I do not see how Pincock can provide can pro-

vide an explanation of how a multiscale representation is partially faithful and

useful. His metaphysical agnosticism position appears to prevent any suitable

structural relationship being established between the target system and the

content of the representation. This is a major problem. It appears that, for

the damped harmonic oscillator and Bénard cell examples at least, Pincock’s

account reduces to the claim that representations are useful because they are

successful, and because they are successful they are partially (in a limited

sense) faithful. But the metaphysical agnosticism blocks (metaphysical) ex-

planations for why they are faithful, so the only explanation for why they are

useful is because they are successful (which is no explanation at all as this is

clearly circular). A possible solution to this problem is that the general point

Pincock is trying to make, that the success of multiscale representations does

not ensure we can gain novel metaphysical features from them, is more of a

guide than a hard and fast rule. That is, in some cases we can gain novel meta-

physical features, in some cases we have to wait, and in some cases they are

explicitly denied. This would mean that the relationship between faithfulness

14Pincock refers to McGivern’s mentioning of fast scale gravity waves and slow-scaleRossby waves (McGivern 2008, pg. 70).

15Including a “wait and see” attitude would imply that we could move to novel metaphys-ical features, we would just have to wait.

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and usefulness will depend on the example at hand; that there is no general

relationship between these two features when multiscale representations are

involved. If this is the case, and this point generalises to any representation,

then Pincock’s account is not so much a unified framework but rather a set of

tools to be adopted and adapted to each case study. He would in fact have

adopted a different methodological approach to Contessa and the proponents

of the IC, where one has a unified framework and attempt to show how exam-

ples fit in with it. This will shift the debate from whether the IC or the PMA

provides the best account of faithful epistemic representation, to whether the

methodological approach of the IC or the PMA is best for accommodating the

ideas behind faithful epistemic representation.

Before switching to this discussion of methodologies, there is one final op-

tion for understanding Pincock in a way that might be compatible with the

unified framework view. This is to understand Pincock as adopting a simi-

lar position as Batterman, that such representations establish a new level of

description of the system. When Pincock discusses the rainbow, he offers up

his account as an alternative, though, similar view to Batterman (Pincock

2012, §11.6). Thus the rainbow can be used to analyse Pincock’s views. In

particular, the β → ∞ idealisation is described to have similar interpretative

challenges as the boundary layer theory, and the CAM approach is at the heart

of the discussion of how similar Pincock’s view is to Batterman.

7.3 The PMA Applied to the Rainbow

7.3.1 β →∞

Pincock advocates the taking of the β → ∞ idealisation over the k → ∞ (or

λ → 0) idealisation due to a desire to provide what he considers the “best”

explanation of the colour distribution of the rainbow available from a ray

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representation (Pincock 2012, pg. 226):

[T]he best explanation of [the colour distribution] takes light to be made

up of electromagnetic waves. This is the best way to make sense of the

colors of light and how the colors come to arrange themselves in the

characteristic pattern displayed by the rainbow . . . Without some rela-

tionship to our wave representation of light, [the ray theoretic] account

of [the rainbow angle] seems to float free of anything we should take

seriously. This point raises an instance of the central question of this

chapter: how can we combine the resources of the clearly incorrect ray

representation with the correct wave representation to provide our best

explanations of features of the rainbow?

The β →∞ and k →∞ idealisations are possible ways of relating the ray and

wave representations. I have already established in §6.3.1 that the β → ∞

idealisation involves a singular limit. I will now look at whether this limit can

be understood in terms of singular perturbation. I will outline claims in favour

of and against understanding the β →∞ idealisation as involving singular per-

turbation and conclude that the β →∞ idealisation does not involve singular

perturbation. I will then argue that this idealisation, and comments Pincock

makes in comparison to the k →∞ idealisation and boundary layer theory, call

into question Pincock’s general approach to interpreting idealisations. At this

stage, it is only a worry. The CAM approach will then be addressed to see

whether it can provide more information on Pincock’s metaphysical agnosti-

cism and its relationship to Batterman’s position.

The most obvious reason that one might think the β → ∞ idealisation

makes use of multiple sets of scales or singular perturbation theory is that the

approach is justified in the same way as scales are. Pincock claims that the

ray representation that results from the β →∞ idealisation “need not require

the assumption that light is a ray. It may assume only that in certain cir-

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cumstances some features of light can be accurately represented using the ray

representation” and that it “can be reliably used if there is a size of raindrop

above which the wave-theoretic aspects of light are not relevant to the path

of the light through the drop” (Pincock 2012, pg. 226, 227). Compare this to

the first discussion of scales for the deep water waves example: there the claim

is that “terms preceded by h become orders of magnitude more important” or

“terms preceded by 1h = λ

H become orders of magnitude less important” (Pin-

cock 2012, pg. 103). In both examples, the the justification of taking β → ∞

idealisation appears to be the same as in the deep water case: a dimensionless

parameter is taken to such a size that other terms do not contribute to the

solution compared to those preceded by the parameter or vice versa.

Secondly, Pincock compares the k → ∞ idealisation with the application

of singular perturbation in the boundary layer theory example, and draws a

lesson from the comparison (Pincock 2012, pg. 227):

we need to pay attention to both the mathematical links between the two

representations and their proper physical interpretation. Some mathe-

matical links preclude any viable interpretation. This fits with the way

singular perturbation theory was used to develop the boundary layer

theory representation in chapter 5 [see §7.2.2 - PMA, Faithfulness and

Usefulness above]. There we saw that the width of the boundary layer

posed an interpretive challenge based on its central place in the repre-

sentation.

I think a fair inference to draw here is that the limit involved in the β → ∞

idealisation is one of these particular mathematical links that requires us to

pay attention to the mathematics and the interpretations, possibly preventing

a viable interpretation. That is, the limit is one that involves singular per-

turbation, and due to being a physical scale, as in the boundary layer case, it

might require careful physical interpretation or even preclude physical inter-

224

pretation. However, precluding a physical interpretation would be odd, as that

is clearly the problem with the k → ∞ idealisation. Thus, one might expect

the β → ∞ idealisation to provide a challenge to being interpreted physical,

but a challenge that should be able to be met.

The claims against the β →∞ idealisation involving singular perturbation

are far stronger than the two claims above. First, the size parameter, β, is

not introduced as a scale parameter, but rather as a standard abbreviating

substitution (replacing several terms with one term). For example, look to

the derivation of the Mie solution in Grandy Jr (2005). The Mie solution is

derived for a homogeneous dielectric sphere, during which equations represent

internal and external electric fields, such as the following equation, (3.83a) in

(Grandy Jr 2005, pg. 86):

Πext1 = icos(φ)

kN0

∞∑l=1

εlalh(1)l (N0kr)P 1

l (cos θ) (7.3)

β is then defined as follows:

β ≡ N0ka = N02πa

λ0

(7.4)

where r = a for the boundary of the water drop. In order to be introduced as a

scale parameter, a suitable variable is supposed to be multiplied or divided by

either k or a, and this clearly has not happened. Second, if the size parameter

were to involve singular perturbation then one would expect there to be some

discussion of the use of perturbation theory in the physics texts. This is not

the case; in fact there is no mention of perturbations in any of the relevant

sections of the physics texts I have used.16

16Perturbation theory is mentioned in the following books in the following contexts, allof which are not relevant to obtaining the ray representation from the Mie solution via theβ → ∞ idealisation: Grandy Jr (2005): inhomogenity (pg. 274) or distorted (i.e. non-spherical) spheres (pg. 297); Liou (2002) contains nothing in the relevant chapter (chapter5); Adam (2002) refers to the topology of caustics being stable under perturbations beingrelevant to rainbows; and Nussenzveig (1992) also discusses perturbations in the context of

225

Third, the β → ∞ idealisation does not result in any features indicative

of singular perturbations. Above I argued that Pincock claimed singular per-

turbations grant us access to novel physical features of systems, but not novel

metaphysical features. The only thing we get out of the β →∞ idealisation is

the claim that we can treat the direction of propagation of the waves as acting

like rays. This is hardly a novel feature, we have used ray optics to describe

the direction light travels for centuries before the Mie solution was obtained.

I therefore conclude that the β → ∞ idealisation does not involve a limit

that can be dealt with by singular perturbation. This conclusion calls into

question Pincock’s comparison to the boundary layer theory and the correct-

ness of the lesson he claims we should learn from the k →∞ idealisation. The

boundary layer theory introduced a difficulty in interpreting the width of the

boundary layer for two reasons: the decomposition of the domain involved

a spatial decomposition; and that there were multiple contradictory ways of

defining the width of the boundary layer. Neither of these features play a

role in the β → ∞ idealisation: not only are there no sets of scales involved,

meaning that there is no actual decomposition of the domain in spatial terms,

there is only one way to define the size parameter. So the problems that arose

with the boundary layer theory are not relevant to the β → ∞ idealisation.

But the comparison seems even more odd when we have a look at what Berry

has to say about the k →∞ idealisation in the passage quoted by Batterman

(Batterman 2001, pg. 88). Berry claims that “[the] shortwave (or, in quan-

tum mechanics, semiclassical) approximation . . . shows that wave functions ψ

are non-analytic in k as k → ∞, so that shortwave approximations cannot be

expressed as a series of powers of 1k , i.e. deviations from the shortwave limit

cannot be obtained by perturbation theory” (Berry 1981, pg. 519-520, emphasis

mine). So neither the β → ∞ nor k → ∞ idealisations involve perturbations.

rainbows as being diffraction caustics in §10.6, as well as the CAM theory of the ripple in§14.4.

226

This muddies the waters over what sort of mathematics Pincock thinks pre-

cludes viable interpretation. It initially appeared that singular perturbations

where the mathematics appeared to be used in an instrumental sense (as in the

boundary layer theory) precluded viable interpretation, and that the k → ∞

idealisation was an idealisation of this type. According to Berry, however, the

k →∞ idealisation is not of this type. So either Pincock’s point is general, in

that it is a warning that any mathematical link might preclude viable inter-

pretation, in which case it is rather vacuous: of course that might happen, and

warning that it might is of no help whatsoever. Or Pincock’s point is supposed

to be about certain mathematical links and that Pincock thought the k →∞

and β → ∞ idealisations are examples of these links and he is mistaken. In

either case, the β → ∞ idealisation does not provide us with any insight into

Pincock’s metaphysical agnosticism as it is not a case of singular perturba-

tion, and so metaphysical agnosticism is not a viable interpretative position

to adopt in response to the idealisation. Whether this dilemma for Pincock’s

lesson is a result of the k →∞ and β →∞ idealisations will now be tested by

turning to the CAM approach and looking at whether it can shed any light on

Pincock’s metaphysical agnosticism.

7.3.2 CAM Approach

The CAM approach promises to be a useful example. Pincock’s interpreta-

tion of the CAM approach can be analysed to learn more about metaphysical

agnosticism by asking the following type of questions. Does he adopt the

metaphysical agnosticism position towards the CAM approach? What are the

reasons for adopting the position he does towards the CAM approach? What

do these reasons tell us about metaphysical agnosticism? It will also allow

the distinction between Pincock’s metaphysical agnosticism and Batterman’s

position to become clear (as this is the use Pincock puts it to himself, as well

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as contrasting his view with that of Belot (2005)). Further, the soundness of

Pincock’s claims of how the CAM approach has to be interpreted can be ques-

tioned by comparing his interpretation with the surplus structure approach

adopted by the IC. Specifically, that the ray theoretic concepts are essential

for the use of the model. In the previous chapter, I argued that the ray theo-

retic concepts were heavily involved in the construction of the model, but are

not required for any serious use of the model (except perhaps for teaching it).

Accepting these arguments would make Pincock’s claims unsound with respect

to using the model to, for example, explain the existence of the supernumerary

bows. Given that Pincock employes the CAM approach within a discussion of

explanation, such a result threatens the soundness of Pincock’s account. This

problem will be developed a little further, while a possible escape is provided

by, again, the possibility that Pincock is following a different methodology,

such that this is not the problem it seems to be.

Pincock’s Interpretation of the CAM Approach

Pincock interprets the saddle points in a ray theoretic way (Pincock 2012, pg.

234). He justifies this interpretation by appealing to scale-like reasoning, that

the saddle points describe a situation where . . .

the dominant contributions . . . arise from electromagnetic waves which

approach the behavior of rays of light. In particular, we have assumed

throughout that we are operating in a context where the wavelength of

light is much smaller than the radius of the drop, the only other relevant

length parameter. So in terms of our size parameter β >> 1.

As the major contribution to the integral comes from the critical points, the

saddle points are conjectured to be associated to rays. Pincock does emphasise

the “experimental nature of this association”, pointing out that the Mie rep-

resentation does not include light rays “in its scope”, i.e. it does not contain

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ray theoretic concepts or make reference to rays. Similarly, the association

between rays and saddle points is not dependent upon the ray theory. Pin-

cock puts this down to our rejection of the ray theory and that ray theory

has nothing to say about saddle points. Pincock has a similar line towards

the other critical points, the Regge-Debye poles (Pincock 2012, pg. 235). He

again points to the association being conjectural, though rather than linking

the poles immediately to surface rays or waves, he initially associates it with

light that has “traveled along the surface of the drop for some distance”. He

argues that we are aware that the ray theory cannot be capturing all of the

light that contributes to the rainbow, in particular the way in which light is

diffracted by spheres, and so we conjecture that the poles are associated with

surface waves. It is unclear whether Pincock intends for the poles to be as-

sociated only with surface waves, or whether he thinks that they should be

associated with surface rays, rays that hit the drop at the critical angle and

travel along the surface of the drop.17

While these associations might be conjectural and yet provide accurate

predictions, Pincock offers other reasons for why the CAM approach can be

considered a good explanation of the rainbow, though I will only discuss the

first of these.18 He argues that each step in the derivation is susceptible to a

physical interpretation (Pincock 2012, pg. 236). This a very different position

to the one that the IC adopts, where the CAM approach is considered surplus

structure, and that (roughly) most steps in the derivation can be considered

to be part of the derivation step. The IC might be capable of adopting a view

similar to this if ‘intermediary’ moves back to empirical set ups are viable. I

do not think that such moves are viable however.19

17See Grandy Jr (2005), pg. 175, and Figure 5.8 on pg. 176. These rays are describedin terms of surface waves here, though given Pincock does not go into details, there is anambiguity as to what he is referring to in the passage on pg. 235.

18The second concerns understanding the CAM approach as an abstract varying repre-sentation.

19For example, if we consider the CAM approach to be surplus structure we might have

229

Pincock asks what the explanations based on the β → ∞ idealisation and

CAM approach mandate for our beliefs about the rainbow. He appeals to

inference to the best explanation pointed towards the mathematical links in-

volved in the explanation, rather than it’s typical use of deducing the existence

of entities (Pincock 2012, pg. 237). With respect to the CAM approach and

the explanation of the supernumerary bows, he argues that (Pincock 2012, pg.

238):

it is not possible to explain the existence and spacing of the supernu-

meraries using just concepts deployed in the ray theory. This is because

interference and diffraction are central to what is being explained. At

the same time, it is not possible to explain this phenomenon by appeal-

ing only to what we find in the wave theory. Important links were made

between the critical points and aspects of the rainbow. These links were

given in terms of rays and surface waves. They involved aspects of the

light scattering that are not available from the perspective of the wave

theory alone. Instead, a scientist must ascend from the wave theory to

the ray representation before she is able to get the ‘physical insight’ into

the supernumeraries that CAM provides. This does not mean that she

must believe that the ray theory is correct. Instead, she must use the

results of one idealization to inform the proper interpretation of another.

The techniques deployed in the explanation of [the rainbow angle] and

the following arrangement: the Mie solution is the empirical set up; the Debye expansionis Model 1; the shift to the complex plane via the Poisson sum formula gives us Model 2;and the result of the CAM approach is the resultant structure of Model 2. This would haveto be mapped via an interpretation mapping to Model 1, and the appropriate structure ofModel 1 to an empirical set up to obtain an interpretation.To obtain a physical interpretation of each step, we might start with the Mie solution as

our empirical set up and consider the Debye expansion to be Model 1. The manipulationof the expansion to a form where it can be transformed by the Poisson sum formula wouldthen be the resultant structure, which we map to an empirical set up. We would then haveto start again with this empirical set up to obtain the CAM approach as a new Model 1.There is a major problem with adopting a physical interpretation of each step in the

above way, namely that it would effectively result in the rejection of the first type of surplusstructure. The statements made throughout the literature on partial structures literatureand the SV concerning this type of surplus structure indicate that it is not a type that oneshould give up. Therefore a version of the IC which includes interpretations for each step ofa derivation should be rejected.

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[the colour distribution] and their proper interpretation let us explain

[the supernumerary bows].

As I argued in §6.3.2, however, Pincock is mistaken about the impossibility

of explaining the rainbow through the use of wave theoretic concepts alone.

Both types of critical points can be associated with wave behaviour: the saddle

points with points of stationary phase and the Regge-Debye poles with surface

waves. The problem for the IC came from the apparent requirement of ray

theoretic concepts in the construction of the model. Pincock’s exposition of

the rainbow and the relevance of the ray theoretic concepts to this, is within

this model creation context. He is concerned with relating each stage of the

derivation to a physical interpretation, and discusses the motivations behind

each move, such as the adoption of the Debye expansion (Pincock 2012, pg.

231, 236). One might judge Pincock’s account to give a better explanation

of the way in which we construct models and how mathematics represents in

such cases, i.e. it has an interpretation during model construction which the

IC does not allow. This might be the case, but would require a comparison

between Pincock’s account and a further developed partial structures version

of the SV, with the IC as a third axis. In terms of model use, I maintain that

Pincock has misunderstood the physics.

I have outlined how Pincock interprets the various aspects of the CAM ap-

proach above. Now I will go on to explore how this relates to his metaphysical

agnosticism and whether it can bring any clarity to this position.

The Rainbow and Metaphysical Agnosticism

In order to gain a proper understanding of metaphysical agnosticism, three

issues need to be settled:

(a) Does the CAM approach involve singular perturbation?

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(b) Given the answer to (a), what does the CAM approach tell us about

metaphysical agnosticism?

(c) How different from Batterman’s position is metaphysical agnosticism?

Issue (a) is obviously the most important. Metaphysical agnosticism is a pos-

sible response to representations that involve singular perturbations, where it

is unclear whether we should adopt an emergentist or reductionist line (pre-

suming that we are rejecting an instrumentalist line).

Singular perturbations involve domains being split either physically, as in

the boundary layer case, or temporally, as in the Bénard cells and damped

harmonic motion cases. The temporal split can be interpreted as being due to

the presence of two processes, one working at a short time scale and one at a

long time scale. Such a split would be inappropriate for the CAM approach;

there are no time variables in any of the relevant equations (e.g. the Mie

solution or the third Debye expansion). Further, both Pincock and Batterman

talk about the rainbow in terms of it providing explanations of the stability of

the rainbow, a use of the representation that would be impossible unless the

representation was timeless. The spatial split is more promising, given that the

use of the ray theory explanation of the rainbow angle and colour distribution is

accepted by Pincock (when arrived at due to the β →∞ idealisation) when the

“wave theoretic aspects of light are not relevant to the path of the light through

the drop” (Pincock 2012, pg. 227). A spatial split of the domain is not required,

however. In the boundary layer theory case, we required singular perturbation

theory as assuming the initial set of scales for the Euler equation resulted in an

empirically false result, namely that the object would experience no drag. None

of the problems with the various rainbow representations were of this kind.

The ray theoretic representation was rejected for not being able to provide an

explanation of the supernumerary bows because it lacked the necessary wave

theoretic features. The Mie solution converged too slowly, which was remedied

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by the adoption of the CAM approach. There is no motivation here for a

spatial split of the domain. There is clearly no use of singular perturbation in

the CAM approach.

Despite the negative answer to (a), the CAM approach is still a strong can-

didate for being interpreted in terms of metaphysical agnosticism. This can

be demonstrated by showing that what Pincock says about instrumentalism

and the metaphysical positions with respect to the CAM approach leaves logi-

cal room for adopting metaphysical agnosticism. Pincock explicitly rejects an

instrumentalist understanding of the mathematics involved in the CAM ap-

proach (or at least explicitly rejects an instrumentalist interpretation of similar

asymptotical moves and therefore by implication those of the CAM approach).

He agrees with Batterman’s claim that (Batterman 1997, pg. 396):

these ‘methods’ of approximation [the asymptotic analysis of the Airy

function] are in effect more than just an instrument for solving an equa-

tion. Rather, they, themselves, make explicit relevant structural features

of the physical situation which are so deeply encoded in the exact equa-

tion as to be hidden from view.

That is, he agrees that the mathematics “can contribute explanatory power”

(Pincock 2012, pg. 239).

Given the agreement between Batterman and Pincock in the explanatory

role these features can play, one can ask whether he also agrees with Bat-

terman’s position towards the relevant structures of the CAM approach (the

Regge-Debye poles and the saddle points). Batterman can often be regarded

as arguing for emergence, as in his (2001). With respect to these emergent

properties and explanations, Batterman makes claims such as the following.

When discussing the caustic-theoretic representation of the rainbow he claims

(Batterman 2001, pg. 96):

It seems reasonable to consider these asymptotically emergent struc-

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tures to constitute the ontology of an explanatory ‘theory’, the charac-

terization of which depends essentially on asymptotic analysis and the

interpretation of the results.

When discussing emergent properties more generally he claims (Batterman

2001, pg. 127):

The explanatory role of the emergents: Emergent properties figure in

novel explanatory stories. These stories involve novel asymptotic theo-

ries irreducible to the fundamental theories of the phenomena.

Other than quoting Batterman’s position on emergent properties, Pincock

makes no mention of understanding the rainbow in reductionist or emergen-

tist terms. He does, however, signal a rejection of a reductionist approach

by reminding the reader of the scaling techniques and how they prevent the

adoption of a reductionist position. This signal, with his attempt at clarifying

the idea behind Batterman’s position, indicates to me that there is room to

reject a metaphysical stance towards the rainbow, and adopt the metaphysical

agnostic position (that is not to say that Pincock does this).

What, then, can the CAM approach tell us about metaphysical agnosti-

cism? The structures we are investigating here are the Regge-Debye poles and

the saddle points (the critical points) and more generally the ray theoretic

structures. We need to get clear on what position Pincock adopts towards

these structures, and how similar these positions are to the metaphysical ag-

nosticism position. Lets start with the rays. When Pincock discusses the rays

in the context of the CAM approach and the explanation of the supernumerary

bows, he again uses talk of a scale like nature: “all we have come to accept as a

result of our explanations is that in certain contexts there are aspects of light

that are accurately captured by the ray representation” (Pincock 2012, pg.

241). He argues that the explanatory power of the ray representation comes

from how it is grounded in the wave theory, i.e. the more fundamental theory.

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In order to understand this point, we need to remember the distinction Pin-

cock draws between theories, representations and models (Pincock 2012, pg.

25-26):

Theory A theory for some domain is a “collection of claims” that aim

“to describe the basic constituents of the domain and how they interact”.

Model Any entity that is used to represent a target system [i.e. a

representational vehicle]. We “use our knowledge of the model to draw

conclusions about the target system”. They may be concrete entities, or

mathematical structures.

Representation A model with content.

Pincock is attempting to argue that we are only ontologically committed to

the wave theoretic concepts we use, as this is the only theory we are entertain-

ing. We can entertain and make use of various representations without being

committed to what they would posit as the constituents of the domain if they

were taken to be theories. This distinction also allows us to take represen-

tations to be explanatorily useful. In this sense, we can use rays to explain

the rainbow, by relating rays to saddle points and Regge-Debye poles, without

being ontologically committed to the rays. We can draw a parallel here to the

way in which the Bénard cells are explained using multiple scales. i.e. The

Bénard cells are considered to be physical features of the system, but we are

not committed to them in any kind of metaphysical way (regarding them as

emergent structures or that they can be reduced). In this situation, the Bénard

cells are analogous to the critical points, and the multiple scales (representing

two different processes) are analogous to the ray theoretic and wave theoretic

contributions to the CAM approach.

One last position needs to be ruled out before we can claim that we can

adopt a metaphysical agnosticism position towards the CAM approach, namely

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the “wait and see” kind of agnosticism (i.e. McGivern’s position). Pincock ar-

gued against a “wait and see” type of agnosticism in the case of the Bénard

cells by rejecting McGivern’s position. I also argued that one could not adopt

a “wait and see” position while maintaining that one was truly rejecting the

reductionist and emergentist positions. Another part of Pincock’s rejection of

the metaphysical positions was that our knowledge of the Bénard cells and the

system through them “may be a product of our ignorance and in the end may

not correspond to what a fuller understanding of the system would reveal”

(Pincock 2012, pg. 119). This is, in part, due to “our limited access to the de-

tails of the fluid dynamics”. This suggests that we can avoid the metaphysical

agnostic position (and possibly embrace one of the metaphysical positions) if

we have a proper understanding of the fundamental theory. This is the case

with the fundamental theory for the CAM approach. Pincock’s argument is

not that we have to use ray theoretic concepts to relate the critical points to

the world due to an ignorance of the wave theory, but that the ray theoretic

concepts are required for relating the critical points to the world. That is,

the only way of connecting the critical points is through ray theoretic con-

cepts. This suggests that Pincock does not hold there to be any metaphysical

difficulties with understanding the rainbow due to ignorance. If there is any

difficulty, it is for other reasons. This further suggests that Pincock does not

adopt metaphysical agnosticism towards the rainbow. The conclusion of the

above argument is that Pincock will adopt some kind of emergentist or reduc-

tionist position towards the critical points. Which position Pincock adopts will

be established in the next section, §7.3.2 - Pincock, Batterman & Belot. Before

I move on to this discussion, a complication with the above argument needs

to be dealt with, which concerns the analogy between the CAM approach and

the Bénard cells.

The idea that our knowledge of the fundamental theory should dictate

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which metaphysical position we adopt leads to a disanalogy between the CAM

approach and the Bénard cells. We have a fundamental theory in each case:

the wave theory for the CAM approach, and the Navier-Stokes theory for

fluids. This raises the question as to why there appears to be no metaphysical

interpretive issues in the case of the CAM approach due to ignorance, but there

are for the Bénard cells. The difference seems to be that we have an explanation

for how the critical points, ray theoretic concepts and wave theoretic concepts

are related in the case of the CAM approach through the use of the theory,

representation and model distinction. In the case of the Bénard cells we do not

appear to have a similar explanation available. The explanation in terms of

the distinction allows us bridge the explanatory gap between the faithfulness

and usefulness in the CAM case. Our ability to judge the (partial) faithfulness

of the CAM representation seems to come from the relationship between the

ray representation and the wave theory, i.e. the taking of the β → ∞ limit.20

The usefulness of the representation is down to our ability to ‘identify hidden

structure’ through the representation, i.e. structure that was already present

in the wave representation but was otherwise inaccessible (Pincock 2012, pg.

221).

An alternative approach to this issue is to ask whether we can consider the

Bénard cells to be ‘hidden structure’ in the way the critical points are. There is

room for this position. Batterman uses the term ‘hidden structure’ to describe

the relevant structures of the rainbow he holds to be emergent. If this position

can be maintained for any structures that might be classed as emergent, then

it is a suitable description for the Bénard cells. If we can class the cells as

structures of this kind, then we can ask the question why we cannot employ

20There is a problem if this is indeed Pincock’s position, as we do not in fact use the thislimit when using the CAM approach; rather we only need β to be greater than 50. Pincockneeds to clarify how essential the actual taking of the limit is to the relationship between theray representation and concepts to the critical points, or whether it is only the possibilityof the limit is which is important.

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the theory, representation and model distinction in the Bénard cells case, i.e.

why we could not consider the Bénard cells to be a representation with no

ontological commitment in the way the critical points are. A possible reason

why the distinction is not relevant is the move to schematic content required

by the use of the singular perturbations. The CAM approach involved shifts to

schematic content when moving into the complex plane, however, so the mere

presence of schematic content cannot be the reason for the difference. The

only viable option seems to be the the supposed ignorance of the fundamental

theory, but it is not clear how we are ignorant of the Navier-Stokes theory.

Pincock’s metaphysical agnosticism is starting to look unjustified. The

above comparison of the rainbow and the Bénard cells suggests that there

is little difference between the two cases, at least insufficient differences to

justify the adoption of a ‘third table’ between the instrumentalist table and

the metaphysical table (reductionism or emergentism). All that remains is

to establish whether there is any difference between the position Pincock does

hold towards the rainbow and Batterman’s, and whether this has any influence

on how we should evaluate metaphysical agnosticism.

Pincock, Batterman & Belot

As identified above, Pincock agrees with Batterman’s claims that the CAM ap-

proach reveals hidden structure and that such structures can be explanatory

(Pincock 2012, pg. 221). The first difference between their positions comes

from Pincock’s rejection of Batterman’s claim that we should adopt the emer-

gent structures as a new ‘theory’ (Batterman 2001, pg. 96):

It seems reasonable to consider these asymptotically emergent struc-

tures to constitute the ontology of an explanatory ‘theory’, the charac-

terization of which depends essentially on asymptotic analysis and the

interpretation of the results.

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To which Pincock responds that “both sympathizers and critics have not agreed

on what Batterman takes the interpretative significance of this ‘theory’ to be”

and subsequently puts forward his distinction between theory, representation

and model (Pincock 2012, pg. 222). It this is novel distinction and attitude

towards representations derived from theories that sets Pincock apart from

Batterman and Belot. Pincock blames the “traditional assumption” that using

a representation derived from a theory requires one to accept the ontology

of that theory to be behind Batterman’s stronger, emergentist position and

Belot’s “deflationary” view.

For instance, Batterman is quoted by Pincock as endorsing a ray theoretic

ontology due to ray theoretic boundary conditions (Batterman 2005b, pg. 159):

those initial and boundary conditions are not devoid of physical content.

They are ‘theory laden’. And, the theory required to characterize them

as appropriate for the rainbow problem in the first place is the theory of

geometrical optics. The so-called ‘pure’ mathematical theory of partial

differential equations is not only motivated by physical interpretation,

but even more, one cannot begin to suggest the appropriate boundary

conditions in a given problem without appeal to a physical interpreta-

tion. In this case, and in others, such suggestions come from an idealized

limiting older (or emeritus) theory.

Pincock’s objection to this is that in using the ray representation (distinguished

from the ray theory as he does), we have not been required to adopt any

believes about light that the ray theory incorrectly offers, whereas Batterman

is endorsing the adoption of such beliefs.

Pincock understands Belot’s position to be a “deflationary” one. Pincock

quotes Belot’s discussion of the relationship between the wave (the more fun-

damental) theory and the ray (the less fundamental) theory (Belot 2005, pg.

151):

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The mathematics of the less fundamental theory is definable in terms

of that of the more fundamental theory; so the requisite mathematical

results can be proved by someone whose repertoire of interpreted phys-

ical theories includes only the latter; and it is far from obvious that the

physical interpretation of such results requires that the mathematics of

the less fundamental theory be given a physical interpretation.

Pincock argues that a correct interpretation of Belot would be to claim that

the success of the CAM approach and catastrophe theory does not entail com-

mitment to the ray theory’s ontology. Pincock also argues, however, that this

“deflationary” reading does not do justice to the explanatory power of the ray

theoretic concepts, in particular the fact that the significance of the critical

points could only be accessed through the use of ray theoretic concepts. Pin-

cock also points to Redhead’s (2004) response to Batterman here as being

deflationary. Redhead agrees with Batterman over the new explanatory power

that can be obtained through asymptotic reasoning, but asks why we cannot

accept these structures as being cases of surplus structure, rather than reify-

ing them and claiming them to be emergent properties or structures (Redhead

2004, pg. 529-530). This indicates that Pincock understands surplus structure

to be an instrumental use of mathematics, a position I rejected in §7.2.2 -

Metaphysical Agnosticism above.

The rejection of the traditional assumption in favour of Pincock’s distinc-

tion between theories and representations does not result in a large difference

between his and Batterman’s views, at least according to Pincock himself. He

believes his . . .

conclusion is completely in the spirit of Batterman’s many remarks on

the interpretive implications of asymptotic reasoning. So I do not intend

my discussion here as a criticism of his views. At the most, what I argue

is that Batterman’s views are not as clear as they should be, and this has

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hampered the appreciation of the significance of cases like the rainbow.

In support of Pincock’s claim that his position is in the spirit of Batterman’s,

Pincock points to a quotation which he claims outline’s Batterman’s “consid-

ered view”: “asymptotic explanation essentially involves reference to idealized

structures such as rays and families of rays, but does not require that we take

such structures to exist” (Batterman 2005a, pg. 162).

It is not my job to argue for whether Batterman’s strong position or what

Pincock considers to be his considered view is correct. I am concerned with

attempting to understanding Pincock’s metaphysical agnosticism, what his at-

titude towards the rainbow is, and whether these positions are distinct from

Batterman’s position. The above discussion makes it clear that what does all

of the work for Pincock is his distinction between theories, representations and

models. This distinction allows Pincock’s attitude towards the rainbow to be

clearly distinguished from Batterman’s strong position: there is no endorse-

ment of any emergent properties, nor any ray ontology, whereas Batterman

at times endorses both of these positions. I take a consequence of Pincock’s

distinction to be a rejection of emergentism, and an endorsement of reduction.

He has effectively argued that, although we need to represent the rainbow with

ray representations, we are not committed to any ray theoretic or emergentist

(as with Batterman) ontologies, which constitutes a rejection of emergentism

and endorsement of reductionism.

The above discussion and model, representation and theory distinction do

little for helping to clear up the mystery around metaphysical agnosticism.

The question can still be raised why we cannot take advantage of the theory,

representation and model distinction towards the Bénard cells. What use is

adopting the metaphysical agnosticism in this situation? Why could we not

describe the situation as requiring a higher level representation, which does not

commit us to any higher level ontology than that of the fundamental theory,

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the Navier-Stokes theory? Although Pincock does talk of the discovery of

novel metaphysical features, I do not see the difference between this and the

supernumerary bows. I do not think that Pincock can produce an answer

to these questions, and therefore reject his metaphysical agnosticism as being

justified.

At best, metaphysical agnosticism is too underdeveloped a position to be

able to hold until more work is done; or at worst, it is untenable, collapsing

into a rough form of reductionism.

7.4 Conclusion

In this chapter I have argued that different structural resources are required on

the PMA for Galilean idealisation and for singular perturbation idealisation.

For Galilean idealisation, the PMA can employ the notions of the structural

relation, which include mathematics, the specification relation and schematic

content. Singular perturbation idealisations are held to have wholly schematic

content at a time, with some representations obtaining an interpretation in

terms of new physical concepts, which requires a new specification relation and

limited structural relation. The issue of how faithfulness and usefulness are

related for each idealisation was not definitively settled in either case. For the

Galilean idealisations, the faithfulness was accounted for through the structural

similarity between target and vehicle, demonstrated through the structural

relation. These idealisations were held to be useful due to the specification

relation, providing interpretations and altering the structural relation to the

specific requirements of the representation at hand. I urged some caution,

here, as these idealisations could be described as being useful ‘because we

design them to be’.

Establishing how faithfulness and usefulness are related for singular pertur-

bation idealisations turned out to be far more difficult. For the singular limits

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that resulted in instrumental type representations, such as the boundary layer

theory, the relationship appeared to be broken. While there were mathemati-

cal links between the target and the vehicle, these were not strictly part of the

representational content for the PMA, and the mathematics that was part of

the content was schematic, i.e. had no interpretation. This schematic content

threatened the possibility of calling the mathematics an epistemic representa-

tion. The introduction of the genuine content provided a possible solution here,

but this move does not provide much help in establishing the link between the

faithfulness and usefulness of the representation with genuine content and the

target. Pincock advocated adopting the position of ‘metaphysical agnosticism’

towards some singular perturbation idealisations, such as the Bénard cells and

damped harmonic oscillators. I argued that this could be understood as a

‘third table’ to the instrumentalist and metaphysical (reductionism or emer-

gentism) tables. Pincock’s original exposition of this position was unclear. I

attempted to gain a clearer understanding of this position through the β →∞

idealisation and the CAM approach. This investigation resulted in a rejection

of the metaphysical agnosticism as a viable position, with the argument that

Pincock should adopt the reductionist position that results from his theory,

representation and model distinction.

I posited that a reason for the difficulties in relating the faithfulness and

usefulness for both types of idealisation might be the difference in methodology

Pincock has in comparison to Contessa and the proponents of the IC. I will

survey this difference in methodology in the next chapter. The next chapter

also involves subjecting the IC and the PMA to the questions set out in Chapter

2, with the aim of choosing between the two accounts.

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8 | Choosing An Account

8.1 Introduction

In Chapter 4 I argued in favour of understanding the IC and the PMA as

accounts of faithful epistemic representation. At the end of the chapter, one

question remained unanswered: question (2.b), which asked “how does it [the

putative account of faithful epistemic representation] account for the relation-

ship between faithfulness and usefulness?” The aim of the last two chapters was

to attempt to answer this question according to each account. I approached

the question by recasting it as the following two questions:

1. What is the relationship between faithfulness and usefulness?

2. What structural resources are used to account for idealisations, and are

they the same across different types of idealisations?

The rainbow was chosen as a case study, to test the answers to the above two

questions. The β → ∞ idealisation and the CAM approach were leveraged

to this end. In this section I will summarise how both accounts were found

to answer the above two questions, and the success or otherwise, of those

answers after being subjected to the β →∞ idealisation and CAM approach.

I will be attempting to draw out more general points than were made in the

previous chapters, so that the lessons learnt there can be brought into the

broader discussion over what a successful account of mathematical scientific

representation should look like. In the next section, §8.2, I will reintroduce

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the problems of the applicability of mathematics, and attempt to answer them

on the IC and the PMA. The questions can be mostly answered, apart from

the problem of misrepresentation due idealisation and abstraction on the PMA.

The solution to this problem was supposed to come out of the discussion in the

previous two chapters, as it did for the IC. In §7.2.2 -Metaphysical Agnosticism

and §7.4 I suggested that the cause of this issue was the methodology Pincock

adopts to construct his account. I will investigate the methodologies and argue

against Pincock’s in §8.3.

8.1.1 The Inferential Conception

The IC provided a good account of the relationship between the faithfulness

of a representation and its usefulness through the notion of partial truth and

the partial structures framework. Idealisations can be understood as ‘as if’ de-

scriptions: descriptions that are treated ‘as if’ they are true in the appropriate

theoretical contexts. This notion of being treated ‘as if’ they were true can

be handled by partial truth, while a measure of how partially faithful a repre-

sentation is can be gained from the notion of how partial a partial structure

is (i.e. the ‘size’ of Ri of the vehicle compared to the ‘size’ of the Ri of the

target). This is an approach that works very well for Galilean idealisations,

while a threat was thought to come from singular limit idealisations. The issue

originated in attempting to understand such idealisations as surplus structure,

a position proponents of the IC hold (Bueno & French 2012, pg. 91).

Much useful work was done in investigating the notion of surplus structure,

such as the splitting of the notion into four roles and assigning the individual

roles to particular parts of the SV and IC. A problem was found with under-

standing singular limits as idealisations. As idealisations on the IC are to be

understood as ‘as if’ descriptions, one could only have an idealisation once one

has a theoretical context, something unobtainable during the derivation step.

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This meant that singular limits should not be understood as idealisations until

the end of the interpretation step.1

Such limits were thought to introduce surplus structure during the immer-

sion stage. I argued that the limits would be surplus structure of the first

type, that which would never have a physical interpretation. A consequence of

adopting partial structures in the derivation in order to accommodate surplus

structure of this type causes one to face the following dilemma:

Very Large R3 Problem the partial structure in the derivation step

must contain all mathematical structures in the R3 component (in case

any mathematical structure is required to be adopted as surplus struc-

ture in the future).

Mathematical Restriction Problem it is unclear what can moti-

vate a restriction in what mathematical moves we hold as possible

moves, and hence limit what is placed in R3 (as the derivation step

takes place within the (uninterpreted) mathematical domain, no physi-

cal considerations can be appealed to).

Possible solutions to both horns of the dilemma were investigated, but all

require work. I’ll cover the most promising response to each horn here.

Solutions offered up to the Mathematical Restrictions Problem were split

into two types: those that are ‘internal’ to the mathematics, i.e. rely solely

on features of the mathematics at hand, and those that are ‘external’ to the

mathematics. The most promising internal solution was to appeal to a no-

tion of there being natural extension of the mathematics included in the first

model, and that only the mathematics that constitutes this natural extension

be included in the R3. Support for this approach came from the analysis of

the CAM approach: I argued that one only needed the typical mathematical

resources that come with transforming a real function into the complex plane1See page 162 for the discussion of this point.

246

and the Euler identity, and that if anything was a candidate for a natural ex-

tension, it is the move into the complex plane for real functions. I also argued

that the Euler identity was a similar example of a natural extension. However,

more work is needed to generalise this solution; it is unclear whether a general

notion of ‘natural extension’ can be explicated that is not too restrictive in

some cases and too permissive in others (though even these notions require

specification).

The solution to the Very Large R3 Problem was to bite the bullet, to admit

that one is committed to all other mathematical structures being ‘in’ the R3.

The most promising approach to this was to argue that the partial structures

are representational devices and so there is only the appearance of a problem

(as being representational devices, there is no problematic commitment in

play). One reason for this approach being the most promising will be developed

below: recognising that the IC is operating at the ‘meta-level’ allows one to

make use of the partial structure framework in a way that does not require

reifying the partial structures. Working at this meta-level allows one to avoid

taking the representation relation, partial structures in the derivation step,

and so on as literally set theoretic relations (i.e. as partial morphisms).

A solution to the dilemma is required for the IC to be held as a viable

account of representation. For the sake of argument, I will assume that a

solution can be found. There are still problems for the IC, however, brought

out by the β → ∞ idealisation. This idealisation demonstrated that not all

singular limits introduce surplus structure. Recognising this required the IC

to be adjusted as to when one could say surplus structure was introduced via

a singular limit. This is not a destructive result for the IC, but the alterations

prompted by the β idealisation caused slight worries.2

While the CAM approach was found to be a great example of surplus

2Specifically, the idea that that we could take a singular limit from an empirical set upseemed to be ‘too quick’. I attempted to assuage these worries. See §6.3.1 for details.

247

structure, it also introduced a further problem. Proponents of the partial

structures framework often talk about its ability to handle both model creation

and use. The CAM approach, and Pincock’s analysis of it, threaten this unified

approach, however, for the specific way in which partial structures are used

in the IC. It appeared that there were good reasons for doubting that the

IC would be able to accommodate the application of mathematics in model

creation. A rough sketch of a solution was offered, the key idea being that the

IC is a third axis in the SV, where there is a ‘horizontal’ axis for inter-theory

relations (such as model creation) and a ‘vertical’ axis between abstract models

and data models (French 2000, pg. 106).

8.1.2 Pincock’s Mapping Account

For the PMA, it was found that different structural resources were required for

different types of idealisations. This resulted in a different relationship between

faithfulness and usefulness for different types of idealisations. For simple and

Galilean idealisations, the PMA provided a good account of both the content

of the representations and of the relationship between the faithfulness and

the usefulness of representations. The notion of enriched content is used to

account for simple idealisations such as the use of real numbers to represent

magnitudes, or regions of space involved in the representation of temperature.

Galilean idealisations are handled by adopting the notion of schematic content,

where part of the mathematics is decoupled from its interpretation. These are

then explained to be partially faithful due to the structural similarity between

the enriched and genuine contents, related by the structural relation. The

usefulness of these idealisations is accounted for by the specification relation,

which provides the interpretation of the mathematics. One can be reductive

and claim that these idealisations are useful because we design them to be.

A problem arose, however, when I attempted to understand the relationship

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between faithfulness and usefulness for representations that involved singular

limits, which were cast as involving singular perturbations by Pincock. In

general, these idealisations are handled by all content being schematic con-

tent, which is then, in part, given a new interpretation, i.e. take part in a

new specification relation. The first example of the boundary layer theory did

not pose a problem by itself, though it was challenging to understand how

it could produce successful predictions on Pincock’s account. Pincock adopts

an instrumentalist line towards the representation, which would indicate that

the content remained entirely schematic. Thus, the boundary layer theory

demonstrated the unusual position Pincock has adopted towards content and

prediction, with predictions not being part of the content of a representation.

The motivation for adopting this instrumentalist position was that the singu-

lar perturbation involved a physical split of the domain, which is physically

unmotivated.

The Bénard cells and damped harmonic motion examples motivated Pin-

cock’s adoption of metaphysical agnosticism. He rejected the instrumentalist

and metaphysical (i.e. reductionism and emergenticism) positions as inter-

pretations of the singular perturbations. I was unconvinced by Pincock’s ar-

guments against the instrumentalist position and his arguments in favour of

rejecting the metaphysical options. I concluded this first analysis of meta-

physical agnosticism with the claim that there is explanatory gap between the

partial faithfulness and usefulness of these representations left by the refusal

to provide any sort of metaphysical, or at least some explicit structural, link

between the target system and the representation. A result of this explanatory

gap is that circularity threatens: the representations are useful because they

provide accurate predictions, and because they provide accurate predictions,

we take them to be partially faithful, and so can take them to be useful.

In an attempt to get clearer on what metaphysical agnosticism is, I turned

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to the rainbow, and the β → ∞ idealisation and the CAM approach again.

Unfortunately my analysis of the β →∞ idealisation provided no clarification,

and only served to damage the PMA: Pincock’s warning that mathematical

links might preclude interpretations is either vacuous (because of being too

general) or is as a result of misunderstanding the physics. Discussion of the

CAM approach could again be split over the issues of model creation and use.

However, little could be said that was of benefit to understanding metaphysical

agnosticism even when this distinction is made. The PMA might well be a

more plausible account of how model creation occurs than one might able to

find on the SV and IC, but I disagree with Pincock over the physics in the

use case. The CAM approach was useful for gaining a better understanding

of metaphysical agnosticism: Pincock makes use of it to contrast his view

to those of Batterman and Belot. My analysis of this discussion lead me to

conclude that metaphysical agnosticism either requires further work to be a

viable position, or that it collapses into a type of reductionist position.

The rejection of metaphysical agnosticism left a hole in Pincock’s account:

the relationship between faithfulness and usefulness was still unexplained. At

the end of §7.2.2 - Metaphysical Agnosticism an attempt of how to save Pin-

cock’s account was sketched. Pincock is not providing a unified framework,

but rather a set of tools to be adopted and adapted towards each case study.

Thus, the position one holds will depend on the example at hand, in a more

relativised way than one would in following the IC. I proposed that this was

due to a methodological difference between the IC and the PMA. This is an

idea that will be pursued in the rest of this chapter. The viability of Pin-

cock’s account will depend on whether the problems outlined in Chapter 7 can

be solved given the results of the investigation into the methodology he has

adopted.

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Before the methodological discussion, I will show how each account in turn

can answer the questions set out in Chapter 2.

8.2 Are the Accounts of Representation

Successful?

The last few chapters were concerned with identifying the internal problems

and issues with the structural accounts of representation. There are ques-

tions these accounts have to be able to answer that are external, namely the

questions and problems raised in Chapter 2. We are now in a position to see

whether the Inferential Conception and Pincock’s Mapping Account are ca-

pable of answering these questions. The applied metaphysical question, the

problem of relating the mathematical domain to the world, was solved in §2.4

where I argued in favour of structural relations. However, as emphasised by

Pincock’s analysis of these questions, the applied metaphysical problem set a

very low bar for its answers. I argued further that specifying exactly what

what was required of the relation fell under the problems of representation.

A structural relation was chosen in part because it could serve as (part of) a

representation relation, hence the endorsement of structural relation accounts

of representation in Chapter 3.

Two problems are straightforward to answer on the IC: the multiple inter-

pretations problem and the isolated mathematics problem. I think that the

IC can also offer a straightforward answer to the novel predictions problem,

though some argumentation is required to establish that the answer is natu-

ralistically acceptable. Similarly, the PMA is capable of providing straight-

forward answers to the multiple interpretations and the isolated mathematics

problems. I also think that the PMA can be extended to answer the novel

predictions problem (Pincock (2012) does not discuss any problems related to

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novel predictions). Whether the answer that the PMA supplies can be consid-

ered a naturalistically acceptable one also requires discussion. This discussion

over (naturalistic) methodology is not to be confused with the discussion over

the problems of representation. In the novel prediction case, the methodology

at stake is that of naturalism, specifically the ‘strict’ naturalism Bangu (2008)

identifies as being incompatible with his Reification Principle. In the case of

the problems of representation, in particular the relationship between faithful-

ness and success, the methodology under discussion is that of the philosophy

of science: what role should philosophy of science, examples, and so on, be

playing in our analysis of science. I will survey the answers to the problems

of misrepresentation I have already given (in §3.3.1 - The Argument from Mis-

representation and §3.5), before discussing the methodological approaches of

Pincock and the advocates of the IC, as the answers to the final questions (mis-

representation due to idealisation and abstractions, and what the relationship

between the faithfulness and usefulness of a representation is) depend on the

outcome of the methodological debate.

8.2.1 The Isolated Mathematics Problem

The SU(3) example shows that a detailed and complete history of the math-

ematics is required to establish whether the mathematics can be truly held to

be isolated. If it is the case that the mathematics is isolated, then a (presum-

ably) unique explanation of why it is applicable must be found. This is not

the case. Both the IC and PMA can straightforwardly answer this problem in

a general way.

The IC and PMA attribute the applicability of mathematics to the struc-

tural relations between the physical and mathematical domains. All that is

required of mathematics is that it provides a structure that is sufficiently sim-

ilar to the physical structure it is applied to (depending on issues related to

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representation). The origins of the mathematics are irrelevant to this way

of understanding applicability and representation. How a mathematician or a

scientist comes to find or construct a piece of mathematics is an interesting his-

torical question, the answer to which has no bearing on the structural resources

provided by the mathematics. On this account of representation, the isolated

mathematics problem is fact a non-problem: the ‘isolation’ of the mathematics

is irrelevant, and so there is nothing relevant about the mathematics to cause

a problem for these accounts.

8.2.2 Multiple Interpretations

The multiple interpretations problem was motivated by the very different ways

in which the negative energy solutions of the Dirac equation were interpreted

by Dirac. They were first rejected as being non-physical, then interpreted as

‘holes’ in a sea of negative energy electrons, and finally as positrons. The

problem asked how these different physical interpretations could have any pre-

dictive success. The start of an answer can be found when the authors of the

IC discuss how one might understand the prediction of the positron on the IC:

(Bueno & Colyvan 2011, pg. 365):

This example [the prediction of the positron] also illustrates how the

inferential conception explains which parts of the mathematical models

refer and which do not. First, the conception provides a framework to

locate and conceptualize the issue: the work is ultimately done at the

interpretation step. Some interpretations are empirically inadequate,

and thus fail to provide an entirely successful account of the applica-

tion process. Dirac’s interpretation of the negative energy solutions as

‘holes’ clearly illustrates this point. However, despite being at best only

partially successful, such empirically inadequate interpretations can be

very helpful in paving the way for empirically successful interpretations.

They offer some understanding of how the world could be if the interpre-

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tation were true, and they can lead the way to interpretations that are

empirically supported. Again, Dirac’s interpretation of the negative en-

ergy solutions in terms of the ‘positron’ beautifully illustrates this point.

As a result, the inferential conception sheds light on the issue of how

mathematical models with non-referring elements can be useful.

Remember that on the IC an interpretation is a partial mapping from the

mathematical structure to a (partial) empirical set up structure. While each

interpretation will require a different partial mapping to differently (partially)

structured empirical set ups, the partial success of the different interpretations

can be explained through there being partial mappings between the empirical

set ups. That is, the structure of the empirical set up that gives the ‘hole’

interpretation might be partially isomorphic to the structure of the empirical

set up that gives the positron interpretation.

The PMA can provide a similar story, though obviously cannot take advan-

tage of the partial structures framework. Here the interpretation comes from

the specification relation. Thus different interpretations of the same mathe-

matics will require different specification relations. There is a slight compli-

cation, however, as Pincock might not hold all of the mathematics involved

in the derivation of the negative energy solutions as being part of the repre-

sentational content. Assume that the negative energy solutions themselves, as

predictions, are not part of the content of the representation. One cannot then

reinterpret the negative energy solutions by relating them to the target via dif-

ferent specification relations because they are not part of the representational

content and thus not actually related by the specification relation. Rather,

we have to relate the original content by a different specification relation (and

therefore possibly a different structural relation) and then follow the steps

through the extrinsic mathematics to get the same solutions. These solutions

will now have a new interpretation due to following from the differently in-

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terpreted (due to the new specification relation) representational content. We

can then talk about the similarities between the specification and structural

relations as being responsible for the success of the different interpretations.

The solutions to the multiple interpretation problems on both accounts

show that the problem arises not due to differences in the mathematics, but

rather in properly relating that mathematics to the world. The solutions talk

about similarities between the structures of the interpreted mathematics, the

empirical set ups, or the target of the specification and structural relations,

rather than about the mathematics itself. The idea that can be pushed here

is that the mathematics is describing the correct structure, but that a physi-

cal understanding of the structure has not been obtained yet. Thus progress

is slowed, because the correct physical consequences cannot be deduced. We

might be able to predict certain behaviours or values via an incorrect interpre-

tation, such as the negative energy solutions describing a positively charged

particle’s movement in an electric field correctly when interpreted in terms of

‘holes’. But this is due to the mathematics representing the correct physical

structure (in part), rather than the interpretation. The ‘hole’ interpretation

was rejected due to physical considerations, such as attempting to understand

it as a proton, which would require the mass of an electron and the proton to

be the same. The underlying idea to the solutions, then, is that the mathemat-

ical structure is correctly describing the world, and thus a Formalist reading

of the mathematics is being used. As such, the solutions to the multiple in-

terpretations problem actually rest upon a solution to the novel predictions

problem.

8.2.3 The Novel Predictions Problem

The novel predictions problem can be set out as follows:

(i) Can we provide an explanation as to why Formalist/Pythagorean pre-

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diction is successful (if one is available)?

(ii) Can we provide a consistent methodological understanding of how novel

predictions are made? (Whether we can have a consistent methodological

treatment of the positron and Ω− predictions, of D-N type and Formal-

ist/Pythagorean type predictions.)

(iii) If we can have such a consistent methodological treatment, can it be one

that conforms with the strict naturalistic view? Or must it be Bangu’s

methodological opportunism, or similar?

(iv) If we cannot have such a consistent methodological treatment, must we

accept Steiner’s pluralism, or a similar position?

In Chapter 2 I surveyed Bangu’s analysis of Steiner’s discussion of prediction

and his, Bangu’s, reconstruction of the argument at the heart of that discus-

sion. I highlighted that Bangu’s naturalist holds there to be no explanation

of why the Reification Principle (RP) succeeds or fails.3 This being the case,

the naturalist would answer negatively to (i). I think that the IC and the

PMA can provide an explanation for why the RP succeeds in some instances

and fails in others. The answer is actually one of the responses that Bangu

argued fails, namely that what is doing the work is the interpretation of the

mathematics.

The interpretation response as set out by Bangu begins by claiming that

“the crucial reificatory step . . . was taken by interpreting the mathematical

formalism, [in that] it is not the formalism itself that predicts, as it were, but

our interpretation of it: . . . without a physical interpretation, no empirical

predictions could ever be obtained from any formalism” (Bangu 2008, pg. 255-

256). This much I agree with. We need to physically interpret the mathematics

3Bangu’s (RP) is as follows: “If Γ and Γ′ are elements of the mathematical formalismdescribing a physical context, and Γ′ is formally similar to Γ, then, if Γ has a physicalreferent, Γ′ has a physical referent as well” (Bangu 2008, pg. 248).

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in order to obtain any kind of physical information. The problem with this

response, as Bangu sees it, is that there is a different notion of interpretation

at work in the case of Formalist or Pythagorean prediction than in typical

cases of mathematical representation. He contrasts the interpretation of the

F in F = ma as force with the interpretation in the case of the prediction of

the Ω−. He describes the F interpretation as being synonymous with “spec-

ify[ing]”, targeting which sort of force we are representing (elastic, electrical,

etc.), whereas he describes the Ω− interpretation as “essentially, . . . reifying

a piece of formalism”. He argues that Gell-Mann’s interpretation of the for-

malism “proceeded (consciously) along analogical-pythagorean lines and was

brought to a specific conclusion by using the RP”.

The root of the problem with the interpretation response, according to

Bangu, is that interpretation, for the naturalist, cannot proceed by ‘reifying

mathematics’, due to it resting on an application of the RP which is natu-

ralistically unacceptable. Thus the solutions offered by the IC and the PMA

need to do two things: explain how the RP can be naturalistically accept-

able; and provide consistent ways of interpreting mathematics. Both accounts

satisfy the second of these conditions: all interpretations on the IC are (par-

tial) mappings from pure mathematical structures to the suitably interpreted

structures of the empirical set up. As explained in §6.2.1, the empirical set

ups are physical interpreted mathematical structures, those that compose the

models used in the SV. Similarly, all interpretations on the PMA are due to the

specification relations. Pincock’s variable notions of representational content

complicate matters slightly, in that at times there will be schematic content.

But this complication only concerns which piece of mathematics is interpreted

in a given representation; it does not effect what interpretation consists in on

the PMA.

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The IC and the PMA can give the same general answer for how the RP can

be successful or unsuccessful by leveraging their answer to the applied meta-

physical question and constitutive question for representation: the structural

relations. The structural relations will relate a part of the physical structure of

the target to a part of a large mathematical structure. The idea is that when

the RP succeeds, the physical structure used as the target of the representation

and the mathematical structure used as the vehicle are similar enough4 that

the derivations and mathematical manipulations we perform ‘reveal more’ of

the mathematical structure that can then be interpreted to ‘reveal more’ of

the physical structure. In cases where the RP fails, the two structures would

be insufficiently similar. The idea here is that the mathematical moves ‘re-

veal more’ mathematical structure, but that these moves do not relate to the

further unknown physical structure. What grounds this idea is that we can fol-

low the structure more easily in the mathematics, i.e. we can use mathematics

to draw surrogative inferences, but due to the large size of the mathematical

structures, we can make ‘false’ moves, i.e. follow relations which are available

in the mathematics but not present in the physical structure. We need a suffi-

ciently similar mathematical structure as our representational vehicle in order

to successfully apply the RP.

Where this response comes up short is that we cannot, ahead of time, know

whether a particular interpretation will be successful. This is only a problem if

one retains the idea that interpretation qua RP is different to the interpretation

of F = ma, but there is no difference between these interpretations on the IC

and PMA. Thus there is no special problem of explaining why ‘Pythagorean’

prediction fails, as it fails for the same reason and traditional prediction fails.

With this answer to (i), we can provide a positive answer to (ii) as well:

both cases of prediction of the positron and the Ω− can be consistently ac-

4In the IC case, sufficiently partially morphic; in the PMA case, whatever notion ofstructural similarity Pincock is happy to use.

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counted for. On the IC, novel prediction is simply the mapping from a piece of

mathematics to a new empirical set up, where that empirical set up is found to

be empirically adequate and the empirical set up contains a novel particle, be-

haviour, etc., that constitutes the novel prediction. The difference between the

prediction of the positron and the Ω− consists in a difference in the response to

the interpretations of the mathematical structures involved in the predictions.

In the case of the prediction of the positron, we had a mathematical structure

that was incorrectly, partially, interpreted which lead to anomalous behaviour

and a reinterpretation of the mathematics. For the Ω− prediction, the mathe-

matical structure could be explored more quickly than the physical structure in

combination with a very close similarity between the mathematical and physi-

cal structure. Thus the prediction of the Ω− appears to be different because no

verification of the results could be carried out until much later. However, the

process was the same. The mathematics was not ‘reified’, but used to explore

what the physical structure might be. It played the representational role of

allowing for surrogative inferences to be drawn about the target, the world.

When one makes ‘Formalist’ or ‘Pythagorean’ predictions and analogies, one

is not reifying mathematics but exploring the structural relations that might

exist in the physical structure. One is predicting what the larger physical

structure will be like, given the larger mathematical structure. I think that

this answer is compatible with a strict naturalistic approach, and so answer

positively to (iii). As the IC is acceptable to the naturalist, its account of

interpretation should also be acceptable.

On the PMA, we can give the same sort of response in general. We are using

mathematics to obtain surrogative inferences, thus in exploring the mathemat-

ical structure we are exploring the possible physical structure that we related

via the structural relation. Thus novel prediction will involve relating the

mathematics to a physical structure that contains a novel prediction. Things

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are made more complicated, however, when we take into account the various

notions of representational content Pincock employes, and that predictions are

not part of the representational content. The first complication is that this

actually seems to block novel prediction of entities: while predictions might

not be part of the representational content, they get their interpretations from

the specification relation and the enriched and genuine content involved in

the representation. This implies that we cannot actually make predictions of

novel entities, as they will not be part of the initial specification relation. Thus

if we start a representation interpreting the mathematics of the Dirac equa-

tion in terms of ‘holes’, or where negative energy solutions are not physically

meaningful, we cannot predict a positron as there is no mathematics without

an interpretation that we can incorporate as ‘new structure’, where this ‘new

structure’ becomes the positron when interpreted. The response is to employ

schematic content. Something might encourage us to reject the interpretation

that we started with, and adopt a schematic content for a piece of mathe-

matics which we can then either perform further work on, or adopt a new

interpretation of, i.e. adopt a different specification relation.

While this response appears to solve the problem, it raises a second compli-

cation. The situation is the same as occurs during the use of singular pertur-

bations. As outlined in §7.2.2, when one makes use of a singular perturbation,

one has to move to wholly schematic content before adopting a new interpre-

tation of the mathematics. However, this is problematic as it is unclear how

the faithfulness and usefulness of representations are related. The same worry

exists here: what justifies the adoption of the new specification relation?

8.2.4 The Problem of Misrepresentation

In §3.5 I argued that some of the problems of misrepresentation were relevant

and legitimate problems for accounts of scientific representation. This argu-

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ment consisted in a rejection of a unified account of representation and the

adoption of Contessa’s accounts, accepting his notions of epistemic and faith-

ful epistemic representation. An epistemic representation is held to be one

that allows for valid surrogative inferences about the target to be drawn, a

(partially) faithful epistemic representation is one that results in (some) sound

surrogative inferences to be drawn. Thus misrepresentation can be understood

as the drawing of false surrogative inferences about the target. Suárez (and

Frigg) argued against structural relation accounts of representation by claim-

ing that they could not account for cases of misrepresentation. In Suárez’

argument, he distinguished two kinds of misrepresentation, each of which in-

volves two further types of misrepresentation: mistargeting; and inaccuracy.

Mistargeting arises when either a representation’s target does not exist (such

as in the case of the mechanical ether or phlogiston), or where one misidentifies

the representation’s target (such as in Suárez’ example of the Pope Innocent X

painting and one’s friend). I dismissed inaccuracy due to empirical inadequate

results as being a general problem for philosophy of science, and claimed that

it did not need to be addressed specifically by an account of representation.

I also argued that mistargeting due to mistaken targets is caused by agents,

rather than the structural relation, and so was not a problem to be dealt with

by accounts of representation, claiming that recognising the fault lies in the

agents’ intentions and uses of representational vehicles solves the problem.

I argued that one’s response to the problem of mistargeting due to non-

existent targets would depend on what use one was attempting to put one’s

representation. The use will dictate whether the representations are successful

or not. If one is attempting to make inferences about the target phenomenon

that are restricted to the phenomenon’s behaviour, then the problem of non-

existent targets need not be a problem, so long as one obtains (suitably) accu-

rate results. However, if one is attempting to use the representation to obtain

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inferences about the ontology of the world, then misrepresentation due to the

non-existence of a target is a major problem.

The issues of idealisation and abstraction5 were dealt with in §6.2 for the

IC and in §7.2 for the PMA. I argued that the IC could account for both

types of idealisation (Galilean and singular limit) that I investigated. It could

also answer the important question raised by adopting Contessa’s framework,

of explaining how the faithfulness and usefulness of a representation could be

linked. The PMA did not fare as well. I could only establish an answer to

the faithfulness and usefulness question for Galilean idealisation, and this an-

swer was not particularly satisfying (that Galilean idealisations are faithful and

useful because we make them be). In the case of the singular limit idealisa-

tions, recast by Pincock as representations that involve singular perturbations,

I found there to be no clear explanation for the relationship between faithful-

ness and usefulness. I suggested in the previous chapter that this lack of clear

explanation was due to a difference in the methodology adopted by Pincock

compared to Bueno, Colyvan, French and Contessa. I will now turn to dis-

cussing the methodologies, in the hope that this will provide a reason for the

lack of explanation of how faithfulness and usefulness are related on the PMA.

8.3 Methodologies

Above I suggested that the methodology Pincock adopted towards creating

his account was the cause of the issue over how faithfulness and usefulness

are related. To form an argument based on this suggestion, I wish to draw

parallels between Pincock’s approach and that of Brading and Landry in their

debate over structural realism with French. Brading and Landry argue6 that

the partial structures framework is not required to account for the applicability

5I did not address abstraction directly, claiming that what was said for abstraction couldbe adapted to accommodate abstraction.

6Brading & Landry (2006), Landry (2007), Landry (2012).

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of mathematics, in that the mathematical relations themselves are sufficient.

French (2012) responds that the root of this rejection of the partial structures

framework is a fundamental disagreement over what the role of philosophers

of science is. This disagreement consists in what French describes as the re-

jection of a ‘meta-level’, a level ‘above’ scientific theories, which are described

as the ‘object level’. It is at the meta-level that the philosopher of science

operates and employs the partial structures framework as a representational

aid to answer questions concerning the structure of, and relationships between,

scientific theories, the applicability of mathematics, and questions of scientific

realism. After setting out this debate in further detail, based on the most

recent contributions, I will argue that Pincock’s attitude towards the repre-

sentation relation is sufficiently similar to Brading and Landry’s attitude to

the mathematical relations to call into question whether he thinks we should

be working with a meta-level.

8.3.1 The Brading, Landry & French Debate

French defends the ‘meta-level’ analysis and use of the partial structures frame-

work by arguing that Brading and Landry’s arguments are based upon two

problematic positions: a misconstrual of the kinds of activity philosophers of

scientists should engage in; and a failure to distinguish between, for example,

the work done by group structure at the ‘object level’ and the work done by

the set-theoretic approach required by philosophy of science at the ‘meta-level’

(French 2012, pg. 3). That theories and the relationships between them are

identified with set-theoretic structures and relations is a common mistake put

forward as a criticism of the the partial structures framework. As far as French

is concerned (French 2012, pg. 5):

set theory offers an appropriate representational device for the philos-

ophy of science and the structural realist in particular; its use should

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not be taken to imply a particular ontological stance with regard to ei-

ther theories themselves or the nature of the structure ‘in’ the world the

realist takes them to represent

Landry’s central claim is that “debates about, or accounts of, scientific

realism, need neither a metaphysical, logical nor mathematical meta-linguistic

framework” (Landry 2012, pg. 30). She aims to show this by arguing that

(Landry 2012, pg. 33):

mathematical structure itself, and yet not just that of physics, is the

‘crucial objectifying structure for science’, so there is no need to ‘deploy

the resources of logic’ to ‘superadd’ to mathematical structure to give

a structural account of concepts, or to give, what [she explains] as, a

structural presentation of kinds of objects

and to

consider what work mathematical structure does at the ‘object level’

of scientific practice and, contra . . . French . . . forego consideration of

what work logical or mathematical structure does at the ‘meta-level’ of

philosophy of science.

Her argument turns on a distinction between the presentation and representa-

tion of objects (Brading & Landry 2006, pg. 573):

[F]or physical theories; theoretical objects, as kinds of physical objects,

may be presented via the shared structure holding between the theoret-

ical models. However, at the ontological level, a physical theory insofar

as it is successful, must also represent particular physical objects and/or

phenomena and not merely present kinds of physical objects.

The idea behind this distinction is that structural mappings are only capable of

identifying shared structure, and that all that is achievable through identifying

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this shared structure is the identification of what kinds of objects the theory

talks about. What is needed is an account of representation that provides an

account of how the theory can move from talking about kinds of objects to be

about particular objects (Brading & Landry 2006, pg. 576).7

Her argument against a meta-level approach is that “unless we presume

that the world is set-structured, appeals to the meta-linguistic structure of

scientific theories have no ontological bite. That is, there is no global, meta-

level, ‘philosophical’ perspective of the structure of scientific theories from

where [the No Miracles Argument] can be put to use” (Landry 2012, pg. 47).

However, she does believe that we can adopt a “ local, object level, ‘scientific’

perspective of the structure of a particular scientific theory from where [the No

Miracles Argument] can be used to ‘cut nature” ’ (Landry 2012, pg. 48). The

argument for this is in part the rejection of the appeal to partial structures

(due to the claim that they do no work in presenting the structure of, for

example, quantum mechanical structures, or in the representation of quantum

mechanical objects) and in part an explanation of, for example, how group-

theoretic structure and group-theoretic morphisms “do the real work ”. The

idea is that when one expresses quantum mechanics group-theoretically, one

forms a theory that presents the structure of quantum mechanical objects

in group-theoretical terms: “e.g. it presents bosons and fermions as group-

theoretically ‘constituted’ kind of objects” (Landry 2012, pg. 50). French’s

position on how set theory should be used, as a representational device at the

meta-level, suggests that Landry is wrong to think that we have to believe

the world is set-theoretically structured in order to have “ontological bite”

from the set-theoretical structure when it operates at the meta-level. Further

to this, French reminds us that “the structure that the structural realist is

concerned with should not be, and should never have been, construed as ‘pure’

7I will leave the details of these positions to the side, as they concern the debate overstructural realism, a debate I am not interested in in this thesis.

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logico-mathematical structure; it was always intended to be understood as

theoretically informed structure” (French 2012, pg. 6).

The point of contention between French and Landry I want to focus on

is how they each specify the notion of shared structure between the physical

(data or phenomena models) and the mathematical models. For French, this is

specified by the notions of partial structure and partial morphisms. For Landry,

the shared structure is made precise at the local level, that is “the shared

structure can be made appropriately precise via the notion of a morphisms and

the context of scientific practice determines what kind of morphism” (French

2012, pg. 6). French quotes her claim that (Landry 2007, pg. 2):

mathematically speaking, there is no reason for our continuing to as-

sume that structures and/or morphisms are ‘made-up’ of sets. Thus, to

account for the fact that two models share structure we do not have to

specify what models, qua types of set-structures, are. It is enough to say

that, in the context under consideration, there is a morphism between

the two systems, qua mathematical or physical models, that makes pre-

cise the claim that they share the appropriate kind of structure.

This context dependency is brought out clearly in what Landry says about the

application of group theory to quantum mechanics. She argues that the mor-

phisms that are doing the work in connecting the models are group-theoretic,

not set-theoretic (e.g. Lie group transformations), and that (Landry 2007, pg.

10-11):

what does the real work is not the framework of set theory (or even

category theory); it is the group-theoretic morphisms alone that serve

to tells us what the appropriate kind of structure is.

French generalises this point as “the use of the concept of shared structure that

determines the kind of structure and characterises the relevant meaning and

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all the relevant work is done by the contextually defined morphisms” (French

2012, pg. 10).

French’s argument can now be summarised: he concentrates on what counts

as relevant ‘work’ from the above quotation by asking the questions ‘who’s us-

ing?’ and ‘what’s working?’ In the group theory and quantum mechanics

example, it’s clear that the physicists and mathematicians use group theory,

rather than partial set-theoretic structures, in the relevant physical contexts:

“in the context of the quantum revolution, it was group theory, not (par-

tial) set-structures that was effectively doing the (physical, mathematical and

hence object level representational) work” (French 2012, pg. 21). Philosophers

of science use the partial set-theoretic structures to represent theories, their

inter-relationships, etc., in order to analyse the structure of scientific theories,

the inter-relationships between theories, answer questions concerning the ap-

plicability of mathematics (e.g. how are the faithfulness and usefulness of a

representation related?), settle ontological questions, and so on. While this is

a representational activity, as well, the representational target is clearly dif-

ferent. Philosophers of science are operating at the ‘meta-level’, where the

target is theories, rather than the world, as is the case at the object level. Set

theoretic structures are doing the work in this representational activity at the

‘meta-level’. So the disagreement comes from . . . :

the need for a meta-level representational unitary framework (provided

by set-theory, category theory, whatever). Brading and Landry insist

that it is ‘shared structure’ (group-theoretic, in the above case study)

that does all the work but the work that is being done is ‘physical’ (!)

work and while I agree that this is appropriate for physicists, philoso-

phers are doing a different kind of work, that requires a different set of

tools. To insist that this form of work should be dismissed would be a

radical step that would fundamentally revise our conception of what the

philosophy of science is all about.

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French concludes that proper philosophical analysis cannot proceed without

employing the appropriate tools at this meta-level, and that a rejection of this

level turns philosophy of science into something it should not be (French 2012,

pg. 26):

Without some formal framework, set-theoretic or otherwise, that can

act as an appropriate mode of representation at the meta-level, our ac-

count of episodes such as the introduction of group theory into quantum

mechanics would amount to nothing more than a meta-level positivis-

tic recitation of the ‘facts’ at the level of practice. Any concern that

the choice of a set theoretic representation of such an account would

imply that set theory is constitutive of the notion of structure can be

assuaged by insisting on the above distinction between levels and modes

of representation.

The “positivist recitation of the ‘facts” ’ is the claim that the group-theoretic

morphisms account for all of the features of science we are attempting to

explain as philosophers of science. I agree with French’s responses to Brading

and Landry. Philosophers of scientists are operating at a ‘meta-level’ and so

require the appropriate tools to operate at this level.

The debate between French and Brading & Landry is over whether we

need the ‘meta-level’ analysis, and so over whether there is any work to be

done by the set-theoretic structures. Part of Brading and Landry’s rejection

of the meta-level is the argument that the context determined morphisms are

sufficient in explaining how objects are presented. Another is a refusal to

identify (or reduce) these morphisms to any underlying, unifying framework

(either set theory or category theory). These two positions lead Brading &

Landry to argue that applications of group theory to quantum mechanics can

be explained in terms of group-theoretic morphisms, such as the transforma-

tions between the Lie groups. To me, this position is similar to Pincock’s view

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that the structural relation that forms the representation relation is context

dependent, in that it can contain mathematical terms. I will now outline how

I view Pincock’s methodology, looking at the role specific examples play on his

account. I will argue that Pincock does adopt some form of meta-level/object

level distinction, but that the context dependence he embraces is the root of

the difficulties I have found with his account thus far.

8.3.2 Pincock’s Methodological Approach

Pincock holds that mathematical representations obtain their content in three

stages: we fix the mathematical structure used as the representational vehi-

cle (the model); we assign (parts of) this model physical significance through

denotation or reference relations; and a structural relation is given so that

parts of the mathematical structure can be mapped to the target system (Pin-

cock 2012, pg. 257). Pincock says very little about how this structural re-

lation should be understood. When outlining his various notions of content,

he argues that we need more complicated relations than isomorphisms and

homomorphisms (Pincock 2012, pg. 31). This greater complexity was to be

introduced through the introduction of mathematics into the structural rela-

tion. For the temperature diffusion example employed during the introduction

of the notion of enriched contents, this consisted in including an error term

with an isomorphism as the structural relation. More generally, this can be

phrased as the enriched contents being “specified in terms of the aspects of

the mathematical structure”, via the structural relation. When providing his

answer to the applied metaphysical question, Pincock argues the simplicity in

answering the problem allows for numerous answers to be given. He rejects

Steiner’s set-theoretic solution, arguing that his structural relations (those set

out in Pincock (2012), Chapter 2), are a possible solution. Pincock endorses

this solution is one does not identify the relations with sets (Pincock 2012, pg.

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174).8 Finally, when discussing the indispensability of mathematics (and the

agreement of the anti-realist positions of Hellman and Lewis with his version of

the indispensability argument), he again comments on the structural relation

being specified, i.e. including, mathematical terms (Pincock 2012, pg. 197).

Given that enriched contents are required for the majority of representations

(ones which go beyond the simple basic contents which require isomorphisms

between target and vehicle), I take it that nearly all mathematical representa-

tions will involve structural relations that include mathematical terms in some

way.

The existence of mathematics in the structural relation brings Pincock’s

position close to Brading’s and Landry’s. This is because the mathematics

involved in the structural relation will be dependent upon the specific target

of the representation. This is clearly seen in the temperature example, where

the structural relation is held to include an error term in order to accommo-

date temperature being defined over regions, while the real numbers describe

a dense continuum (Pincock 2012, pg. 31). Pincock does not discuss any other

representation in such detailed terms, so it is unclear what he would take to

be included in other cases. One can make an educated guess however. In the

irrotational fluid example (used in his exposition of abstract varying repre-

sentations (Pincock 2012, §4.2)), the Navier-Stokes equation is simplified by

adopting several idealisations: that the fluid is incompressible and homoge-

nous, which results in a constant density; that there is a steady flow of fluid,

which results in the equations being time independent; and that we ignore the

viscosity of the fluid, which involves setting the term that represents viscosity,

µ, to 1 (Pincock 2012, pg. 69). I think it is reasonable to take the structural

relation required for this representation to include the setting of the density to

a constant value and µ = 1. It would also not relate to time, though this leads

8This mirrors Landry’s position of refusing to identify the morphisms with set-theoreticor category-theoretic morphisms (Landry 2007, pg. 2).

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to schematic content.9 In the rainbow example, Pincock talks of the critical

points being connected to the rainbow through both ray and wave theoretic

concepts (Pincock 2012, pg. 238). While I disagree with Pincock’s analysis

of the representation of the rainbow, if we are to accept what Pincock says

here, I think that the structural relation involved would have to include ray

and wave mathematics in order to do its job of relating the CAM model to the

rainbow in the way Pincock claims it is. From these examples and how the

structural relations need to be structured, i.e. what mathematics they need

to include, I can infer how Pincock would analyse Landry’s group-theoretic

quantum mechanics example. It requires either a group-theoretic mapping as

the structural relation in the extreme case, or group-theoretic concepts in the

structural relation in the conservative case.

One might infer from the above argument that I take Pincock to implicitly

reject the need for a meta-level analysis of representation. I do not think that

this is the case. Rather I think that his focus on what he considers to be the

epistemic benefits of mathematics and a focus on case studies has shaped his

account in a way that prevents it from properly answering the question over

how faithfulness and usefulness are related. When outlining his approach,

Pincock is explicit in looking for a “general strategy” (Pincock 2012, pg. 5) for

answering the questions he is looking at (Pincock 2012, pg. 3):

[F]or a given physical situation, context, and mathematical representa-

tion of the situation [we can ask:]

(1) what does the mathematics contribute to the representation,

(2) how does it make this contribution, and

(3) what must be in place for this contribution to occur?

Pincock’s account of representation is therefore aimed at providing general an-9Pincock explains why this is the case when outlining the moves involved in establishing

a ‘steady state’ representation of traffic flow (Pincock 2012, pg. 55).

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swers to these questions, and should be capable of providing general answers

to the questions I raised in Chapter 2. Indeed, he must think his account is

capable of this, as he devotes an entire chapter to analysing Steiner’s questions

(Pincock 2012, §8).10 What, then, has gone wrong for Pincock? In §8.3.3 I

will provide a brief survey of what he takes to be some of the main epistemic

benefits of the use of mathematics in scientific representations. This survey

indicates that all of these can be accounted for by the IC (and thus by ac-

counts of representation that explicitly focus on the surrogative inferences).

So the problem is not caused directly by the different focus on the epistemic

benefits itself. Rather, I think the problem is that the structural relation in-

cludes mathematics specific to the representation at hand. This trivialises

the obtaining of faithful representation, which violates Contessa’s maxim that

while obtaining epistemic representation is cheap, obtaining faithful epistemic

representation is costly.

I will now provide the brief survey of the epistemic contributions and outline

how they can be accommodated through surrogative inferences. Two of these

contributions will be given further space as I take Pincock’s arguments for how

his account accommodates them to be based on a position I do not think he

should hold.

8.3.3 Accounting For Pincock’s Epistemic Contributions

The epistemic contributions Pincock believes mathematics makes to science

include (Pincock 2012, pg. 8):11

i) “aiding in the confirmation of the accuracy of a given representation

through prediction and experimentation”;

10I even relied on some of this analysis in my Chapter 2.11Throughout this thesis I have not discussed the constitutive role (or the derivative role)

of mathematics that Pincock outlines in his Chapter 6, as I do not take it to be relevantto the questions I am concerned with. As such, the epistemic contributions Pincock takesmathematics to contribute due to these roles have been ignored here.

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ii) “calibrating the content of a given representation to the evidence avail-

able”;

iii) “making an otherwise irresolvable problem tractable”;

iv) “offering crucial insights into the nature of physical systems”.

This is in addition to the “metaphysical role of isolating fundamentally math-

ematical structures inherent in the physical world”. ‘Inherent’ here should be

read in a way that is appropriate for one’s account of representation. In the

case of the IC, this would be that the target systems could be structured in

the way required to be considered empirical set ups. In Pincock’s case, this

would be cashed out in terms of the notion of instantiation he adopts (Pincock

2012, pg. 29). Pincock also identifies further epistemic contributions that come

from the various types of representations he identifies (Pincock 2012, pg. 9-11).

Abstract acausal representations offer the following contributions:

(v) if the acausal representation was obtained from a causal representation,

then we might gain information about the system that was not explicit

in the casual representation;

(vi) the acausal representation is easier to confirm, due to having less content.

The abstract varying representations also involve particular epistemic contri-

butions:

vii) enable the transfer of evidential support from one representation (in the

family) to another;

viii) indirect contribution from failures of the representation to treat all mem-

bers of the family equal (Pincock 2012, pg. 82).

Either due to:

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(viii.a) there being more shared structure than identified by the represen-

tation;

(viii.b) overextension demonstrating that a member of the family is differ-

ent to an/the other member/s in some respect.

And finally the representations concerned with scale supposedly provide the

contribution of (Pincock 2012, pg. 119):

ix) a third response to the debate over metaphysical interpretations and

instrumentalism concerning ‘emergent’ properties

I have already looked at the viability of ix) in §7.2.2 through §7.3.2, and the

arguments in this chapter concerning the methodology Pincock has adopted.

Contributions iii), iv) and (v) are all straightforwardly handled through the

ability to draw sound surrogative inferences in our representational vehicle. I

understand the contribution of i) to simply be the obtaining of sound surrog-

ative inferences, that is, of establishing a (partially) faithful representation.

I take the idea behind the notion of “calibrating the content” included in ii)

to be that one aims to include only what one is is aiming to represent in the

world in one’s representation, though this is masked by being set out in a way

that appears rather specific to Pincock’s account. As this will involve finding

suitable mathematics on any account of representation, I do not take this to

be a specific contribution obtained only through Pincock’s account. I take

(vi) to be independent of one’s account of representation: it is a claim about

one’s account of confirmation and how that relates to ‘how much’ content a

representation has. I take that any representation that results in less content

(however that is measured) will be easier to confirm if one adopts a similar

account of confirmation as Pincock does (i.e. a Bayesian inspired account.)12

The contributions identified in (viii.a) and (viii.b) require individual anal-

ysis. These are both due to abstract varying representations. These are rep-12See Pincock (2012, §2.8-2.9.

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resentations that involve a mathematical model that is variously interpreted

in terms of different systems. These systems might be different only in that

they are individual systems of the same type, e.g. two copper springs, or dif-

ferent in nearly every way, such as (in Pincock’s example) irrotational fluids

and electrostatics.

The two contributions in (viii.a) and (viii.b) are held to be due to there

being some shared structure between the two systems (this is the basis behind

there being abstract varying representations in the first place). However, Pin-

cock places more emphasis on this shared structure than I think he should be

able to if he is following his account properly. In particular, given the claims

he makes over how mathematics is interpreted during a representation.

Both cases are held to be indirect contributions due to there being inac-

curacies when applying the abstract varying representations. (viii.a) occurs

when “there is some underlying mathematical similarity to all the systems,

but the current family of representations has missed this in some respects”

(Pincock 2012, pg. 82). (viii.b) occurs when one member of the family can

be represented with an extension of the mathematics, but another member

cannot. This can lead to differences between the members being identified.

The idea behind these contributions is that the targets of the abstract varying

representations share a certain amount of structure that is partially repro-

duced in the representation. These arguments also require that one is able to

discuss the mathematical similarities or dissimilarities between the represen-

tations and draw conclusions from this. Such a discussion requires the ability

to divorce the mathematics from its interpretation (to discuss the mathemati-

cal structure itself) and then hold that the mathematical structure is in some

sense shared/instantiated/etc. (depending on one’s account of representation)

between the targets. But Pincock faces a problem with both of these re-

quirements. First, Pincock can only talk of the content of representations in

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uninterpreted terms when it is shifted to a schematic content. This means

that the abstract varying representations would only be able to provide some

of their epistemic benefits when they are not, in fact, being ‘various inter-

preted’, but when they are shifted to schematic versions that do not have

interpretations. Second, Pincock is clear in outlining these epistemic benefits

as being obtainable when mathematics is not playing a “metaphysical role”,

that is, when it is not being claimed to be shared/instantiated/etc. in the

target system (Pincock 2012, pg. 8). But this second requirement only makes

sense if mathematics is playing the “metaphysical role”: unless one claims that

the targets of the abstract varying structures have some structure in common

that the mathematics is reproducing (as per one’s account of representation),

then the mathematical arguments are unfounded.

These advantages can be accounted for by the IC: we map to the same

mathematical model in each case due to each of the target systems being par-

tially -morphic to the repeatedly used mathematical model. As we are now in

the domain of uninterpreted mathematics, we can make the necessary math-

ematical arguments to derive our conclusions. The interpretation mappings

then let us attempt to relate our mathematical results to the target systems,

and the failure (or not) of these partial mappings can inform the fit of the

mathematical conclusions to the target system. (viii.a) will occur when we

have mapped ‘too small’ a partial structure to the mathematical domain from

the empirical set up, and the same partial mappings are used in the immersion

and interpretation steps in all of the varying representations.13 Thus the same

piece of the target systems’ structures are missed in each application. (viii.b)

occurs when the same partial mapping is used in the immersion step, but the

13By ‘same’ mapping here, I mean a mapping that takes the same mathematical elementsand relations to suitable elements and relations in the empirical structured in the same way.That is, the mappings will result in the same structure in each empirical set up. The differentinterpretation comes from the structures in each empirical set up being of different physicalsystems.

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resultant structure after the ‘over extension’ cannot be mapped to different

empirical set ups using the same interpretation mapping. The idea here is

that we have the same partial structure mapped from the different empirical

set ups to the mathematical domain, but that there is no ‘larger’ partial struc-

ture in common between the empirical set ups. Hence some extension of the

‘small’ mathematical structure to a ‘larger’ mathematical structure will result

in us being unable to use the same interpretation mapping in all instances.

As the IC (and other accounts of representation that focus on surroga-

tive inferences) can accommodate the epistemic contributions outlined in this

section, another argument in favour of Pincock’s account is blocked. If these

types of account could not accommodate the epistemic contributions, then

they would be missing something related to mathematical representation that

Pincock’s account has identified. However, this is not the case. Thus Pincock’s

account does not appear to offer anything over and above what the IC itself is

capable of providing.

I will now, finally, choose between the IC and the PMA as my favoured account

of mathematical representation.

8.4 Choosing An Account

As I highlighted above, the irrotational fluid example included several ideal-

isations (e.g. the fluid is incompressible and homogenous, so the density is

a constant), which would require mathematics specific to that example being

included in the structural relation. I then argued that in more complex cases,

the structural relation would end up being close to the type of relation that

Brading and Landry claim as being sufficient to account for the representa-

tions. This same relation is one that French would understand to be operating

at the ‘object level’. I agreed with French’s argument that the work we are

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required to do as philosophers of science require its own set of tools, and that

set theory is one such tool. It is important to remember is that partial struc-

tures are being used to represent structures and relationships at the object

level in our philosophical work at the meta-level. By including mathematics in

the structural relation in the way I understand Pincock to be, the distinction

between the object level and meta-level is blurred. It becomes unclear how we

can use mathematics (that is, structural relations that include mathematical

terms) to represent the relationship between mathematics and physical sys-

tems to answer questions of how mathematics is applicable, in general, at the

meta-level. A consequence of this is that the work required of the philosopher

of science becomes more difficult to achieve. Hence my difficulty in identifying

the relationship between faithfulness and usefulness in Pincock’s analysis of

the singular perturbation representations. If the structural relations respon-

sible for representational content are specified in the way Pincock requires,

making use of case specific mathematics, then accounting for our ability to

use representational vehicles to draw sound surrogative inferences is a simple

matter of pointing to the structural relation. As I claimed when discussing

Pincock’s analysis of Galilean idealisations, and what now appears to hold for

all representations on Pincock’s account, representations are faithful and use-

ful on Pincock’s account because we design them to be. The idea is that we

design our representations to have a sufficiently close a fit to our target system

such that the results are more than likely to be correct, and so we trivially

obtain faithful representations.

I subscribe to Contessa’s maxim that faithful epistemic representation does

not come cheap (Contessa 2011, pg. 127). His brief argument for this is that it

doesn’t take much to draw surrogative inferences with a representational ve-

hicle about a target (hence epistemic representation being cheap), but it does

take a lot to explain how we are able to construct faithful epistemic representa-

278

tions that are sufficiently faithful for our (scientific) purposes. This is a maxim

made and argued for from the meta-level perspective. If we ask how represen-

tations are able to grant us sound inferences at the object level, the answer

is rather obvious: the representations are physically interpreted mathemat-

ics that are related to the target system through (very specific) mathematical

relationships. But these object level questions can only provide case specific

answers. When operating at the meta-level, we want general answers to our

questions, as the questions at this level are general themselves. Providing ad-

equate answers to questions at this level is difficult, given the range of cases

that such an answer will have to cover, hence the cost involved in providing

a suitable account of faithful epistemic representation. Due to Pincock’s blur-

ring of the object and meta-levels, I take his answers to trivialise this difficulty,

providing answers that are too case specific than is suitable when operating at

the properly distinguished meta-level. For this reason I am inclined to reject

his account as a viable account of representation.

In contrast to Pincock’s Mapping Account, the Inferential Conception has

answered the questions set out in Chapter 2. It is able to account for relation-

ship between the faithfulness and usefulness of representations in a way that

respects the costs involved in answering the question, and does so at what I

take to be the appropriate level (a clearly distinguished meta-level). While the

account still requires some work (see §6.2.3 onwards), it is a more successful

account of representation than Pincock’s.

A final worry can be levelled at the IC. The arguments in this chapter

rely on the meta-level/object level distinction. By adopting this distinction,

I am committing myself to employing the partial structures framework in a

representational capacity. Thus, my answers to the problems of the applicabil-

ity of mathematics are at the meta-level. This role and the meta-level/object

level distinction was employed by French to resist the reductive moves Landry

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wished to make (e.g. that group theory was really set theory, and so set theory

accounts for quantum phenomena). One might therefore take the answers I

have supplied via the IC to not actually answer the constitutive question of

Chapter 3, nor the applied metaphysical question of Chapter 2. If the par-

tial structures framework of the IC is in fact representing relationships at the

object level, then I have not provided an account of what the representation

relation, or the relation responsible for the applicability of mathematics, is at

the object level. And it at this level that the constitution question and applied

metaphysical question are operating. My response to this charge to insist that

these questions operate at the meta-level. The meta-level is simply the level

at which philosophers of science operate. Thus any philosophical question is a

question at the meta-level. Confusing the levels causes the problems found in

Pincock’s account.

8.5 Conclusion

In this chapter I have brought the questions of Chapter 2 to bear on the IC and

the PMA. The both accounts could provide answers to the isolated mathemat-

ics, multiple interpretations and novel predictions problems. The IC provided

a good answer to the problems of representation. The attempt to answer these

questions on the PMA prompted an investigation into the methodology Pin-

cock adopted in constructing his account. The conclusion of this investigation

was that Pincock’s methodology trivialised the obtaining of faithful epistemic

representation. This occurred due to the focus on examples and inclusion of

mathematics in the structural relation. The trivialisation of faithful epistemic

representation in combination with the quality and cohesiveness of the IC’s

answers to the other problems prompted me to choose the IC as the superior

account of mathematical representation.

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I will now draw the thesis to a close by summarising the conclusions of each

chapter and the thesis as a whole in the next chapter. I will also identify where

further work can be pursued, and whether such work can improve the position

of each account.

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9 | Conclusion

In this thesis I set out to answer several of the problems of the applicability

of mathematics. In Chapter 2 I argued that there are nine problems that fall

under this banner, and that the problems of representation were the most sig-

nificant problems. Answers to these problems set limits on the relation that

answers the applied metaphysical problem and form the basis for answering

the problem of isolated mathematics and the novel predictions problem. I

argued in favour of structural accounts of representation in Chapter 3, con-

cluding that they offered the resources necessary to provide accounts of faithful

epistemic representation. A partially faithful representation is one that facil-

itates the drawing of at least one sound surrogative inference from vehicle to

target. I argued in favour of the Inferential Conception of the Applicability

of Mathematics, set out in Chapter 6, throughout the rest of the thesis and

rejected Pincock’s Mapping Account, set out in Chapter 7. I introduced the

case study of the rainbow, concentrating on the β → ∞ idealisation and the

CAM approach, which is required to explain the supernumerary bows. I used

the case study to establish whether each account could explain the relation-

ship between the faithfulness and usefulness of representations. My argument

against Pincock’s Mapping Account relied on a problem with the methodol-

ogy Pincock adopted in constructing his account. His focus on examples and

incorporating mathematics into the structural relation that formed part of his

representation relation trivialised the relationship between the faithfulness and

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usefulness of representations. Contessa’s maxim holds that establishing how

the faithfulness and usefulness of representations are related is a costly exer-

cise. Pincock’s account therefore violates this maxim, and subsequently was

rejected.

According to the Inferential Conception, mathematics is applicable due to

a structural relation between the mathematical and physical domains. This

relation should be understood as a partial mapping between partial structures.

The IC is capable of providing answers to the problems of representation (the

multiple interpretations problem and misrepresentation due to abstraction and

idealisation), isolated mathematics and novel prediction problems. It holds

that whether mathematics is isolated or not is irrelevant to its capacity to be

applied; all that matters is whether it can enter the appropriate structural

relationships with the world. The multiple interpretation problem is answered

by attributing the success of inconsistent interpretations of a piece of mathe-

matics to the structural similarities between the empirical set ups, the targets

of the representation. The novel predictions problem is answered straightfor-

wardly; all interpretation on the IC consists in a partial mapping between the

mathematics and the empirical set up, thus there is no difference between so

called Pythagorean or Formalist predictions and typical examples of predic-

tion. Thus a novel prediction occurs when there is a novel entity, behaviour or

so on in the empirical set up irrespective of the supposedly distinct types of

prediction used to obtain that novelty. How the IC answers the issues related

to misrepresentation due to abstraction and idealisation will be outlined below,

after I outline how the IC might contribute to answering one of the problems

of applicability I did not address in this thesis.

The most relevant problem of applicability to the IC that I did not address

is the mathematical explanation problem: how does mathematics provide gen-

uine explanations of physical phenomena? This has already been touched on

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by proponents of the IC, such as Bueno & French (2012). At minimum, the IC

provides a framework to approach the question. First, it allows us to identify

where the explanation might occur. Within the derivation step, the mathemat-

ics is all uninterpreted, and so any putative mathematical explanation would

be an ‘internal’ mathematical explanation, whereas what is required is an ‘ex-

ternal’ explanation: an explanation of a physical phenomena that is external

to the mathematical domain.1 Thus the explanation would have to come from

some mathematics within the final empirical set up. A strict reading of the IC

would then appear to rule out any mathematical explanation, as the mathe-

matics in the empirical set up is interpreted. i.e. Physical facts would be doing

the explanatory work, rather than mathematical facts. A possible response is

that the mathematics in the R3 component of the empirical set up could pro-

vide a mathematical explanation as it is still uninterpreted. Second, accounts

of explanation that might allow for mathematical explanations would have to

be consistent with the IC. The IC looks to be a promising way to approach

this issue. Hopefully it can also provide a similar framework to approach the

semantic problem.

The IC was found to be a fairly successful account of faithful epistemic

representation. The relationship between the faithfulness and usefulness of

representations could be accommodated by the partial structures framework

and partial truth. A ‘measure’ of the partiality of a representation could

be achieved by comparing the size of the Ri between the vehicle and target.

Representations can be considered useful, however partially faithful they are,

as they can be considered as ‘as if’ descriptions. Here the notion of partial truth

comes into play. Within a theoretical context, provided by the interpretation of

the mathematics, the representation can be considered ‘as if’ it were true. This

requires a revision to when we can claim a piece of mathematics as constituting

1Here I am adopting the notions of internal and external mathematical explanationoutlined by Baker (2009).

284

an idealisation: strictly, there are no idealised representations on the IC, as

the mathematics is uninterpreted while it is a representational vehicle. We

can only call the mathematics an idealisation once it has a theoretical context,

which comes when we interpret it. Thus we have idealised representations

when the empirical set ups are used as the representational vehicles in other

parts of the Semantic View.

Another issue related to this is the notion of surplus structure. I argued that

the fourth role of surplus structure, that of accommodating Hesse’s tripartite

analysis of analogies, should be restricted to the horizontal axis of the SV, while

the second (surplus mathematics that is initially not interpreted, but later

does receive an interpretation) and third roles (the extra mathematics involved

in idealisations such as representing magnitudes by real numbers) should be

restricted to the final empirical set ups, after the interpretation mappings.

The first role of surplus structure, mathematics that never receives a physical

interpretation, should be restricted to the first step of the IC, the immersion

step. I argued that in order to accommodate this role of surplus structure

several changes would have to be made to how singular limit idealisations are

typically addressed and how the ‘models’ of the IC are understood would have

to be altered. First, the SV could no longer be committed to the idea that a

singular limit necessitated a move to surplus structure. This was shown to be

the case through the β →∞ idealisation. Second, surplus structure has to be

understood as a family of structures that provide novel surrogative inferential

relations. A consequence of this is that when we iterate the immersion mapping

to obtain a second model, the surplus structure, this ‘model’ is actually a family

of structures. Third, I generated a dilemma for accommodating the surplus

structure within the R3 component of the partial structures. Briefly, either one

has to be committed to what I named the Very Large R3, i.e. all mathematics

(or structure suitably (partially) -morphic to all mathematics) would have to

285

be included in the R3 component, or one must find some way to restrict what is

included in the R3 component. I sketched several solutions to each horn of this

dilemma. The most promising of these solutions was the ‘natural’ restriction

solution to the Mathematical Restriction Problem. This solution requires one

to include only that mathematics which could be considered a natural extension

to the mathematics one is currently employing. This solution gained support

through my analysis of the CAM approach.

There is still further work to be done on these solutions. All of the potential

solutions for the surplus structure dilemma require further development. For

example, the notion of ‘natural’ central to the natural restriction solution needs

to be explored to ensure that it is not too permissive or too chauvinistic in

other cases. Perhaps the most significant further work that needs to be carried

out is to develop the IC as a third axis to the SV. The SV can be considered

to have a horizontal axis, which consists of inter-theory relationships, and a

vertical axis, which consists of relationships between data, phenomena and

theoretical models (French 2000, pg. 106). The ‘points’ along each of these

axes are the models of the SV, which are the representational targets of the

the IC.2 Each of these models will include mathematics to some degree, and

so require the resources of the IC to explicate how the mathematics represents

in each instance. Exactly how the IC can fit as a third axis with the other

two required development. Issues that need to be investigated include how

the IC fits with the dynamic inter-theory relations along the horizontal axis

and whether the problems of model creation and model use are solved through

adopting the IC as a third axis.

Pincock’s Mapping Account was initially promising. His various notions of

content and more relaxed approach to the structural relations (by including

2The axis metaphor is not to be taken too literally here. A model should not have athree valued co-ordinate. Rather ‘axis’ should be thought of more in terms of dimensionsalong which moves can be made to other models of varying features.

286

mathematics in the relations) appeared to provide a good framework within

which idealisations could be easily accommodated, and the relationship be-

tween the faithfulness and usefulness of the representations could be accounted

for. This initial hope was misplaced however. The answers I obtained from

Pincock’s Mapping Account regarding the issues of how the faithfulness and

usefulness of representations are related were found wanting for both types of

idealisation. Galilean idealisations were found to be faithful due to the struc-

tural similarities between the vehicle and the target, and useful due to the

structural relations. However, the justification for why the structural similar-

ities and relations ground the notions of faithfulness and usefulness appeared

to be based on the notion that we made these idealisations faithful and useful

for our purposes, which does not properly answer the question.

With respect to singular limit idealisations (or as cast by Pincock: singular

perturbation idealisations), Pincock’s account failed in different ways depend-

ing on what sort of singular limit idealisation was involved. Apparently in-

strumental cases, such as the boundary layer, provided no explanation for how

the faithfulness and usefulness of the representation were related. Noting that

the representation has to involve genuine content to obtain a prediction did

not help, as this merely involves specifying parameters for the case at hand;

it does not explain why the idealisations and mathematics actually produce

useful answers. Initially, it seemed that adopting a position of metaphysical ag-

nosticism towards other singular perturbation cases, such as the Bénard cells,

resulted in a circular justification for why these representations were faithful

and useful. I conducted further investigation into the metaphysical agnosti-

cism position by comparing and contrasting it with what Pincock had to say

about the β → ∞ idealisation and CAM approach. The conclusion of this

investigation was that the metaphysical agnosticism position was either unjus-

tified (due to being underdeveloped) or collapses into some kind of reductionist

287

position, given Pincock’s theory, representation and model distinction.

A final attempt to understand what Pincock might say concerning the

relationship between the faithfulness and usefulness of representations was

conducted by assessing the methodology Pincock adopted in constructing his

account. I argued that Pincock’s approach of adopting structural relations

that incorporate mathematics dependent upon the example at hand trivialises

the relationship between faithfulness and usefulness on his account. For this

reason, I feel that he not satisfied Contessa’s maxim that it should be difficult

to obtain faithful epistemic representation, and so rejected his account in favour

of the IC. It is not clear that any adjustments could save Pincock’s account.

Any restriction of the structural relation to the typical set theoretic structural

mappings of iso- or homo- morphisms would restrict one’s ability to adopt

enriched contents. A move to partial structures might be one way to go. The

obvious question to ask in response to this is whether Pincock’s account would

collapse into, or could be merged with, the IC. One reason I think this is

unlikely is that the PMA allows mathematics to be interpreted while acting as

a representational vehicle, while the IC does not. This is an insurmountable

difference.

288

A | Appendix: Mathematics of the

Rainbow

A.1 Introduction

In this appendix I will set out the derivation of the exact Mie solution for the

scattering of an electromagnetic spherical wave by a transparent sphere. This

derivation will be mostly taken from Liou (2002) and Grandy Jr (2005). I

will also set out a large part of the derivation of the Debye expansion of the

Mie solution, taken from Nussenzveig (1969a, 1992). I will briefly mention the

relevant equations for the Poisson sum formula, which is used in extending

real functions to the complex plane, before outlining the important points of

the derivation of the unified CAM approach, in particular the CFU method

for evaluating the contributions of the saddle points, taken from Nussenzveig

(1992). Finally, I will provide a brief summary of mathematics behind poles,

which are the other kind of critical point involved in the CAM approach, the

Regge-Debye poles.

289

A.2 Mie Representation1

We start with Maxwell’s equations, where E is the electric vector, D is the

electric displacement, j is the electric current density, B is the magnetic in-

duction vector, H is the magnetic vector, c is the speed of light, t is time and

ρ is the density of charge:

∇ ×H = 1

c

∂D

∂t+ 4π

cj (A.1)

∇ ×E = −1

c

∂B

∂t(A.2)

∇ ⋅D = 4πρ (A.3)

∇ ⋅B = 0 (A.4)

Since ∇ ⋅∇ ×H = 0, performing a dot product on (A.1) leads to

∇ ⋅ j = − 1

4π∇ ⋅ ∂D

∂t(A.5)

Differentiate (A.3) to get:

∂ρ

∂t+∇ ⋅ j = 0 (A.6)

To get a unique determination of the field vector from a given distribution of

current and charges, the above equations must be supplemented by relation-

ships that describe the behaviour of substances under the influence of the field.

1The majority of the derivation is taken from (Liou 2002, pg. 177-188), and (Grandy Jr2005, pg. 100-101).

290

These are given by:

j = σE (A.7)

D = εE (A.8)

B = µH (A.9)

where σ is the specific conductivity, ε the permittivity and µ the magnetic

permeability.

Consider the situation where there are no charges, ρ = 0, no current, ∣j∣ = 0,

and the medium is homogeneous so that ε and µ are constants. Thus we can

reduce the Maxwell equations to:

∇ ×H = εc

∂E

∂t(A.10)

∇ ×E = −µc

∂H

∂t(A.11)

∇ ⋅E = 0 (A.12)

∇ ⋅H = 0 (A.13)

These equations, (A.10) - (A.13), are used to derive the electromagnetic wave

equation. We consider a plane electromagnetic wave in a periodic field with a

circular frequency ω so that:

E→ Eeiωt (A.14)

H→Heiωt (A.15)

291

This allows (A.10) and (A.11) to be transformed into:

∇ ×H = ikm2E (A.16)

∇ ×E = −ikH (A.17)

where k is the wave number (k = 2πλ = ω

c ), m = √ε is the complex refractive

index of the medium at frequency ω, and µ ≈ 1 is the permeability of air.

We perform the curl operation on (A.17). Then, making use of the vector

identity ∇ × (∇ ×V) = ∇ (∇ ⋅V) − ∇2V and noting that ∇ × ∇ × E = 0 and

∇ ⋅E = 0, and similarly for H, we can obtain:

∇2E = −k2m2E (A.18)

∇2H = −k2m2H (A.19)

From these two equations, we can see that the electric vector and the magnetic

induction in a homogeneous medium satisfy the vector wave equation in the

form of (A.20), where A can be either E or H:

∇2A + k2m2A = 0 (A.20)

If ψ satisfies the scalar wave equation (A.21), the variables of which can be

separated through the definition of (A.23). The scalar wave equation is given

in spherical coordinates in (A.22).

0 = ∇2ψ + k2m2ψ (A.21)

0 = 1

r2

∂

∂r(r2∂φ

∂r) + 1

r2 sin θ

∂

∂θ(sin θ

∂ψ

∂θ) 1

r2 sin θ

∂2ψ

∂φ2+ k2m2ψ (A.22)

ψ (r, θ, φ) = R(r)Θ(θ)Φ(φ) (A.23)

292

we can define vectors Mψ and Nψ which satisfy (A.20). These vectors in

spherical coordinates (r, θ, φ) are:

Mψ = ∇ × [ar (rψ)]

= (ar∂

∂r+ aθ

1

r

∂

∂θ+ aφ

1

r sin θ

∂

∂φ) × [ar (rψ)]

= aθ1

r sin θ

∂ (rψ)∂φ

− aφ1

r

∂ (rψ)∂θ

(A.24)

mkNψ = ∇ ×Mψ

= arr [∂2 (rψ)∂r2

+m2k2 (rψ)] + aθ1

r

∂2 (rψ)∂r∂θ

+ aφ1

r sin θ

∂2 (rψ)∂r∂φ

(A.25)

The terms ar,aθ,aφ are unit vectors in the spherical coordinates.

Using two independent solutions to the scalar wave equation, u and v, we

find electric and magnetic field vectors (A.26) and (A.27) respectively, that sat-

isfy (A.16) and (A.17) respectively. This allows us to write E and H explicitly

as follows in (A.28) and (A.29):

E =Mv + iNu (A.26)

H =m (−Mu + iNv) (A.27)

E = ari

mk[∂

2 (ru)∂r2

+m2k2 (ru)] + aθ [1

r sin θ

∂ (rv)∂φ

+ i

mkr

∂2 (ru)∂r∂θ

]

+ aφ [−1

r

∂ (rv)∂θ

+ 1

mkr sin θ

∂2 (ru)∂r∂φ

] (A.28)

H = ari

k[∂

2 (rv)∂r2

+m2k2 (rv)] + aθ [−m

r sin θ

∂ (ru)∂φ

+ i

kr

∂2 (rv)∂r∂θ

]

+ aφ [m

r

∂ (ru)∂θ

+ i

kr sin θ

∂2 (rv)∂r∂φ

] (A.29)

We separate the variables of the scalar wave equation in spherical coordi-

293

nates by substituting (A.23) into (A.22), then multiply by r2 sin2 θ to obtain:

[sin2 θ1

R

∂

∂r(r2∂R

∂r) + sin θ

1

Θ

∂

∂θ(sin θ

∂Θ

∂θ) + k2m2r2 sin2 θ] + 1

Φ

∂2Φ

∂φ2= 0

(A.30)

Due to the dependence of the first three terms of (A.30) on r and θ, but not

φ, the equation is only valid when the following holds:

1

Φ

∂2Φ

∂φ2= constant = −`2 (A.31)

where we set the constant ` equal to an integer for mathematical convenience.

Substituting (A.31) into (A.30) and dividing through by sin2 θ we can write:

1

R

∂

∂r(r2∂R

∂r) + k2m2r2 + 1

sin θ

1

Θ

∂

∂θ(sin θ

∂Θ

∂θ) − `2

sin2 θ= 0 (A.32)

which requires the following to hold, where n is an integer:

1

R

d

dr(r2dR

dr) + k2m2r2 = const = n(n + 1) (A.33)

1

sin θ

1

Θ

d

dθ(sin θ

dΘ

dθ) − `2

sin2 θ= const = −n(n + 1) (A.34)

Rearranging (A.31), (A.33) and (A.34) leads to:

d2 (rR)dr2

+ [k2m2 − n(n + 1)r2

] (rR) = 0 (A.35)

1

sin θ

d

dθ(sin θ

dΘ

dθ) + [n(n + 1) − `2

sin2 θ] θ = 0 (A.36)

d2Φ

dφ2+ `2φ = 0 (A.37)

294

The single-value solution for (A.37) is:

φ = a` cos `φ + b` sin `φ (A.38)

where a` and b` are arbitrary constants.

(A.36) is the equation for spherical harmonics. By introducing µ = cos θ we

can rewrite (A.36) as:

d

dµ[(1 − µ) dΘ

dµ] + [n(n + 1) − `2

1 − µ2]Θ = 0 (A.39)

so that its solutions can be expressed by the associated Legendre polynomials

(which are spherical harmonics of the first kind) in the form of (A.40).

Θ = P `n (µ) = P `

n (cos θ) (A.40)

We can solve (A.35) by setting the variables kmr = ρ and R = 1√ρZ (ρ) to

obtain (A.41):

d2Z

dρ2+ 1

ρ

dZ

dρ+⎡⎢⎢⎢⎢⎣1 −

(n + 12)2

ρ2

⎤⎥⎥⎥⎥⎦Z = 0 (A.41)

The solutions to this equation can be expressed by the general cylindrical

function of order n + 12 , given by:

Z = Zn+ 12(ρ) (A.42)

Thus the solution of (A.35) is:

R = 1√kmr

Zn+ 12(kmr) (A.43)

Combining the solutions (A.38), (A.40) and (A.43) allows us to express the

295

elementary wave functions at all points on the surface of a sphere as:

ψ (r, θ, φ) = 1√kmr

Zn+ 12(kmr)P `

n (cos θ) (a` cos `φ + b` sin `φ) (A.44)

The cylindrical functions in (A.43) can be expressed as a linear combina-

tion of two cylindrical functions, the Bessel function Jn+ 12and the Neumann

function Nn+ 12. We thus define the Riccati-Bessel functions as:

ψn(ρ) =√πρ

2Jn+ 1

2(ρ) (A.45)

χn(ρ) =√πρ

2Nn+ 1

2(ρ) (A.46)

ψn are regular in every finite domain of the ρ plane including the origin, whereas

χn have singularities at the origin ρ = 0, where they become infinite. Because

of this, we can use ψn to represent the wave inside the scattering sphere, but

not χn. Utilising the Bessel and Neumann functions, (A.43) can be rewritten

as

rR = cnψn (kmr) + dnχn (kmr) (A.47)

where cn and dn are arbitrary constants. (A.47) is the general solution of

(A.35).

The general solution of the scalar wave equation (A.22) can then be ex-

pressed as:

rψ (r, θ, φ) =∞∑n=0

n

∑`=−n

P `n (cos θ) [cnψn (kmr) + dnχn (kmr)] (a` cos `φ + b` sin `φ)

(A.48)

As cn and dn are arbitrary constants, we can set them to cn = 1 and dn = i

296

so that we can write the section of (A.48) in square brackets as follows:

ψn (ρ) + iχn (ρ) =√πρ

2H

(2)n+ 1

2

(ρ) = ξn (ρ) (A.49)

where H(2)n+ 1

2

is the half-integral-order Henkel function of the second kind, which

vanishes at infinity in the complex plane, and is thus suitable for representing

the scattered wave.

The general solution of the scalar wave equation (A.22) can now be ex-

pressed as follows in terms of the spherical Riccati-Bessel and Riccati-Hankel

functions:

rψ (r, θ, φ) =∞∑n=0

n

∑`=−n

P `n (cos θ) ξn (kmr) (a` cos `φ + b` sin `φ) (A.50)

Above, we have solved the vector wave equation in general. We can use the

above analysis to describe the scattering of a plane wave by a homogeneous

sphere. For simplicity we assume that the medium outside of the sphere has a

refractive index of 1 (it is a vacuum), that the sphere has a refractive index of

m and that the incident radiation is linearly polarised. This allows us to chose

a Cartesian coordinate system with its origin at the centre of the sphere, the

positive z axis along the direction of propagation of the incident plane wave.

If one normalises the amplitude of the incident wave to 1, then the incident

electric and magnetic field vectors are given by:

Ei = axe−ikz (A.51)

Hi = aye−ikz (A.52)

where ax and ay are unit vectors along the x and y axes respectively. We can

transform the components of any vector in Cartesian coordinates, e.g. a, to

spherical coordinates. If ar,aθ,aφ are the relevant unit vectors in spherical

297

coordinates, then we can make use of the following geometric identifies to

perform this transformation:

ar = ax sin θ cosφ + ay sin θ sinφ + az cos θ (A.53)

aθ = ax cos θ cosφ + ay cos θ sinφ − az sin θ (A.54)

aφ = −ax sin θ + ay cosφ (A.55)

Thus the electric field vector can be expressed in spherical coordinates as

follows:

Eir = e−ikr cos θ sin θ cosφ (A.56)

Eiθ = e−ikr cos θ cos θ cosφ (A.57)

Eiφ = −e−ikr cos θ sinφ (A.58)

The magnetic field vector can be ignored, as to derive the coefficients that

will be presenting in the scattering functions only one of the components of

either the electric or magnetic field vectors will be required. We will use

the component Eir. Looking to (A.28), we can see that the first term is this

component. Thus (m = 1):

Eir = e−ikr cos θ sin θ cosφ = i

k[∂

2 (rui)∂r2

+ k2 (rui)] (A.59)

This allows us to find the potentials u and v. To do so we make use of the

following identities (A.61)-(A.63) and Bauer’s formula (A.60) (Liou 2002, pg.

298

183):

e−ikr cos θ =∞∑n=0

(−i)n (2n + 1) ψn (kr)kr

Pn (cos θ) (A.60)

e−ikr cos θ sin θ = 1

ikr

∂

∂θ(e−ikr cos θ) (A.61)

∂

∂θPn (cos θ) = −P 1

n (cos θ) (A.62)

P 10 (cos θ) = 0 (A.63)

In light of the above four equations we can write (A.56) as:

e−ikr cos θ sin θ cosφ = 1

(kr)n−1

∞∑n=1

(−i)2 (2n + 1)ψn (kr)P 1n (cos θ) cosφ (A.64)

We take a trial solution of (A.59) by using an expanding series of a similar

(mathematical) form:

rui = 1

k

∞∑n=1

αnψn (kr)P 1n cosφ (A.65)

We can then substitute (A.65) and (A.64) into (A.59), and through a compar-

ison of coefficients we obtain (A.66):

αn [k2ψn (kr) +∂2ψn (kr)

∂r2] = (−i)n (2n + 1) ψn (kr)

r2(A.66)

In order to establish the coefficient αn we must refer back to (A.47). Since we

know that χn (kr) become infinite at the origin (through which the incident

wave must pass), we can set the coefficients (which are arbitrary) as follows:

cn = 1 and dn = 0 (such that the χn do not feature). It follows that:

ψn (kr) = rR (A.67)

As (A.47) is the general solution of (A.35), (A.67) is as well, so long as α =

299

n(n + 1) in (A.68):

d2ψndr2

+ (k2 − α

r2)ψn = 0 (A.68)

Thus, comparing (A.68) and (A.66) we find αn to be

αn = (−i)n 2n + 1

n(n + 1) (A.69)

Similar procedures gives vi (the potential v for the incident plane wave)

from (A.29). Thus for the incident waves outside of the sphere, hence super-

script i, we have the following:

rui = 1

k

∞∑n=1

(−i)n 2n + 1

n(n + 1)ψn (kr)P1n (cos θ) cosφ (A.70)

rvi = 1

k

∞∑n=1

(−i)n 2n + 1

n(n + 1)ψn (kr)P1n (cos θ) sinφ (A.71)

In order to find the scattered wave potentials, we need to match ui and

vi with the potentials for the scattered wave inside and outside of the sphere.

We do this by expressing these potentials in a similar form, but with arbitrary

coefficients. These potentials are derived from (A.50), which was explained

above to be suitable to represent the scattered wave.

For internal scattered waves, i.e. transmitted waves hence superscript t, as

χn (mkr) are infinite at the origin, only the ψn (mkr) feature:

rut = 1

mk

∞∑n=1

(−i)n 2n + 1

n(n + 1)cnψn (mkr)P1n (cos θ) cosφ (A.72)

rvt = 1

mk

∞∑n=1

(−i)n 2n + 1

n(n + 1)dnψn (mkr)P1n (cos θ) sinφ (A.73)

However, for external scattered waves, hence superscript s, the ψn (mkr)

functions must vanish at infinity. Thus we make use of the Hankel functions

300

expressed in (A.49):

rus = 1

k

∞∑n=1

(−i)n 2n + 1

n(n + 1)anξn (kr)P1n (cos θ) cosφ (A.74)

rvs = 1

k

∞∑n=1

(−i)n 2n + 1

n(n + 1)bnξn (kr)P1n (cos θ) sinφ (A.75)

In order to establish what the coefficients an, bn, cn, dn are, we make use of

the boundary conditions at the surface of the sphere: the components of E

and H are continuous across the spherical surface r = a such that:

Eiθ +Es

θ = Etθ (A.76)

Eiφ +Es

φ = Etφ (A.77)

H iθ +Hs

θ =H tθ (A.78)

H iφ +Hs

φ =H tθ (A.79)

Comparing (A.28) and (A.29) to (A.70)-(A.73), and making use of the bound-

ary conditions above, we see that the expressions v and 1m∂(ru)∂r for the electric

field and mu and ∂(rv)∂r for the magnetic field are continuous at the boundary,

r = a. Thus:

∂

∂r[r (ui + vi)] = 1

m

∂

∂r(rut) (A.80)

ui + us =mut (A.81)

∂

∂r[r (vi + vs)] = ∂

∂r(rvt) (A.82)

vi + vs = vt (A.83)

301

From (A.80)-(A.83) and (A.72)-(A.75) we have:

m [ψ′n (ka) − anξ′n (ka)] = cnψ′n (mka) (A.84)

[ψ′n (ka) − bnξ′n (ka)] = dnψ′n (mka) (A.85)

[ψn (ka) − anξn (ka)] = cnψn (mka) (A.86)

m [ψn (ka) − bnξn (ka)] = dnψn (mka) (A.87)

We can eliminate cn and dn to find an and bn in terms of the size parameter

β = ka = 2πλ a:

an =ψ′n(mβ)ψn(β) −mψn(mβ)ψ′n(β)ψ′n(mβ)ξn(β) −mψn(mβ)ξ′n(β)

(A.88)

bn =mψ′n(mβ)ψn(β) − ψn(mβ)ψ′n(β)mψ′n(mβ)ξn(β) − ψn(mβ)ξ′n(β)

(A.89)

cn =m [ψ′n(β)ξn(β) − ψn(β)ξ′n(β)]ψ′n(mβ)ξn(β) −mψn(mβ)ξ′n(β)

(A.90)

dn =m [ψ′n(β)ξn(β) − ψn(β)ξ′n(β)]mψ′n(mβ)ξn(β) − ψn(mβ)ξ′n(β)

(A.91)

In all “practical applications”, observations are made at the “far-field zone”

(Liou 2002, pg. 186). This ‘zone’ allows the introduction of an approximation:

the Hankel functions of (A.49) reduce to the form:

ξn (kr) ≈ in+1e−ikr kr ≫ 1 (A.92)

302

which results in the following approximations:

rus ≈ −ie−ikr cosφ

k

∞∑n=1

2n + 1

n (n + 1)anP1n (cos θ) (A.93)

rvs ≈ −ie−ikr sinφ

k

∞∑n=1

2n + 1

n (n + 1)bnP1n (cos θ) (A.94)

Esr =Hs

r ≈ 0 (A.95)

Esθ =Hs

φ ≈−ikre−ikr cosφ

∞∑n=1

2n + 1

n (n + 1) [andP 1

n (cos θ)dθ

+ bnP 1n (cos θ)sin θ

] (A.96)

Esφ =Hs

θ ≈−ikre−ikr sinφ

∞∑n=1

2n + 1

n (n + 1) [anP 1n (cos θ)sin θ

+ bndP 1

n (cos θ)dθ

] (A.97)

The radial components, Esr and Hs

r can be ignored in the far-field zone. We

can simplify (A.96) and (A.97) through the introduction of two scattering

functions :

S1 (β, θ) =∞∑n=1

2n + 1

n (n + 1) [an(β)πn (cos θ) + bn(β)τn (cos θ)] (A.98)

S2 (β, θ) =∞∑n=1

2n + 1

n (n + 1) [bn(β)πn (cos θ) + an(β)τn (cos θ)] (A.99)

where:

πn (cos θ) = P1n (cos θ)sin θ

(A.100)

τn (cos θ) = dP1n (cos θ)dθ

(A.101)

Thus we may write the simplified versions of (A.96) and (A.97):

Esθ =

i

kre−ikr cosφS2 (θ) (A.102)

−Esφ =

i

kre−ikr sinφS1 (θ) (A.103)

We need to alter the notation of (A.88), (A.89), (A.100), (A.101) to match

303

that used in the latter part of the derivation. The approach to doing this is

taken from Grandy Jr (2005). To do this, we express (A.88), (A.89), (A.90)

and (A.91) via new quantities:

P en ≡ ψn(β)ψ′n(mβ) −mψn(mβ)ψ′n(β) (A.104)

Qen ≡ χn(β)ψ′n(mβ) −mψn(mβ)χ′n(β) (A.105)

Pmn ≡ ψn(mβ)ψ′n(mβ) −mψ′n(mβ)ψ′n(β) (A.106)

Qmn ≡ χ′n(β)ψn(mβ) −mψ′n(mβ)χn(β) (A.107)

where superscript e denotes the electric components and superscript m denotes

the magnetic components. This allows us to express the coefficients as:

an =P en

P en + iQe

n

(A.108)

bn =Pmn

Pmn + iQm

n

(A.109)

cn =−im

P en + iQe

n

(A.110)

dn =im

Pmn + iQm

n

(A.111)

Note that Grandy uses Hankel functions of the first type. The Hankel functions

are complex conjugates of each other: (ξ(1)n (z))∗ = ξ2n(z). I believe this is the

reason for the difference between the coefficients given in Grandy Jr. (e.g.

his bn is equal to −bn from Liou: cf. equation (5.2.74) (Liou 2002, pg. 185)

and equation (3.88b) (Grandy Jr 2005, pg. 87).) Grandy Jr. also makes use

of the Wronskian of the ψn(x) and ξn(x) functions (A.113). This allows the

nominators of cn and dn to be written in terms of i and the complex refractive

304

index, m:

W f(x), g(x) ≡ f(x)g′(x) − f ′(x)g(x) (A.112)

W ψn(z), χn(z) = 1 (A.113)

The Wronskian allows the numerator of cn to be transformed as follows:

m [ψ′n(β)ξn(β) − ψn(β)ξ′n(β)]

=m [ψ′n(β) (ψn(β) + iχn(β)) − ψn(β) (ψ′n(β) + iχ′n(β))]

Dropping the subscript n and (β), rearranging and then by (A.113)

=m [ψψ′ + iψ′χ − ψψ′ − iψχ′]

=m [ψψ′ − ψψ′ + i (ψ′χ − ψχ′)]

=m [−i (ψχ′ − ψ′χ)]

=m [−i (1)]

= −im

This notation allows one to gain “insight into the physical meaning of the

coefficients” (Grandy Jr 2005, pg. 100). Taking m to be real for a moment,

one can define a real phase shift δn as:

tan δen ≡P en

Qen

(A.114)

tan δmn ≡ Pmn

Qmn

(A.115)

305

so that the coefficients are equal to:

an =tan δentan δen

= 1

2(1 − e2iδen) (A.116)

bn =tan δmntan δmn

= 1

2(1 − e2iδmn ) (A.117)

(A.116) can be expanded (and similarly for (A.117)):

an =P e2n

P e2n +Qe2

n

− i P enQ

2en

P e2n +Q2e

n

(A.118)

Grandy Jr. points us to scalar wave scattering and the expression for the

partial scattering amplitude he derived:

fn(k) =e−in

π2

2ik[Sn (ka) − 1] (A.119)

where Sn (ka) ≡ e2iδn is the partial-wave ‘scattering matrix’ in terms of phase

shifts. We can follow a similar path here by defining the matrix:

Sn ≡⎛⎜⎜⎝

SEn 0

0 SMn

⎞⎟⎟⎠

(A.120)

For real m, Sn must be unitary (by conservation of energy). Thus SEn and SMn

must be phase factors, and can only be those introduced above in (A.116) and

(A.117):

an =1

2[1 − SEn (β)] (A.121)

bn =1

2[1 − SMn (β)] (A.122)

Substituting into (A.88) and (A.89) allows us to express (A.121) and (A.122)

306

as:

SEn (β) = −ζ2n (β)ζ1n (β)

( ln′ ζ2n (β) −m−1 ln′ψn (mβ)

ln′ ζ1n (β) −m−1 ln′ψn (mβ)

) (A.123)

SMn (β) = −ζ2n (β)ζ1n (β)

( ln′ ζ2n (β) −m ln′ψn (mβ)

ln′ ζ1n (β) −m ln′ψn (mβ)

) (A.124)

where we have changed to using ζ(1,2) to denote the Riccati-Hankel functions of

type 1 and 2 respectively, and we define the logarithmic derivative as ln f(x) =d ln f(x)dx = f ′(x)

f(x) . For convenience these two types of multipole can be combined

into the single expression where v1 = 1 and v2 =m−2, j = 1,2:

Sjn (β) = −ζ2n (β)ζ1n (β)

( ln′ ζ2n (β) −mvj ln′ψn (mβ)

ln′ ζ1n (β) −mvj ln′ψn (mβ)

) (A.125)

A further modification to the notation of (A.98) and (A.99) has to be made.

The motivation behind this modification is to introduce angular functions,

pn (cos θ) and tn (cos θ), that are “more useful for [analytic] extrapolation to

complex indices” than those found in (A.98) and (A.99) (Grandy Jr 2005,

pg. 133). These angular functions “contain only simple Legendre polynomials,

remove the explicit appearance of numerical factors, and exclude values with

n = 0”. With x = cos θ and for all n ≠ 0 we define:

pn (x) =[Pn−1(x) − Pn+1(x)]

1 − x2= 2n + 1

n (n + 1)πn(x) (A.126)

tn (x) = −xpn (x) + (2n + 1)Pn (x) = 2n + 1

n (n + 1)τn(x) (A.127)

Thus (A.125) can be written as follows:

Sj (β, θ) =1

2

∞∑nS=1

[1 − S(j)n (β)] tn (cos θ) + [1 − S(i)

n (β)]pn (cos θ) (i, j = 1,2i ≠ j)

(A.128)

307

This is the exact Mie solution for the scattering amplitudes.

The dimensionless scattering amplitude can be expressed as a partial-wave

expansion in terms of the scattering amplitudes:

f (k, θ) = 1

ika

∞∑n=0

(n + 1

2) [Sn − 1]Pn (cos θ) (A.129)

A.3 Debye Expansion2

The Debye expansion involves representing the multipole of index n in the par-

tial wave sums (A.123) - (A.125) in terms of incoming and outgoing spherical

waves that are partially transmitted and partially reflected at the surface of

the sphere (Grandy Jr 2005, pg. 144) (Adam 2002, pg. 283). We employ the

Debye expansion as applying the Poisson sum formula, (A.155), to the Mie

solution directly results in residue series that converge no quicker than Mie

solution itself.

We start by introducing the radial wavefunctions for the interior region of

the sphere, denoted as region 1, and the exterior region of the sphere, denoted

as region 2. We can also introduce spherical reflection and transmission coef-

ficients so that we can write the radial wavefunction in each region, for each

partial wave, as:

φ2,n(r) = A⎛⎝ζ(2)n (kr)ζ(1)n (β)

+R22ζ(1)n (kr)ζ(1)n (β)

⎞⎠

(A.130)

φ1,n(r) = A⎛⎝T21

ζ(2)n (mkr)ζ(2)n (mβ)

⎞⎠

(A.131)

where A is a normalisation constant. φ and its derivative, φ′, must be continu-

ous at the surface of the sphere, r = a. This results in the relation T21 = 1+R22

and gives us enough information for us to determine R22 and T21. The Rij

2This derivation is summarised from (Grandy Jr 2005, pg. 144-153) and Nussenzveig(1969a,b).

308

are spherical reflection coefficients and the Tij are spherical transmission coef-

ficients, where i, j = 1,2, denoting the regions the spherical wave is reflecting

from or transmitting to.

An outgoing spherical wave that interacts with the surface is described by:

φ2,n(r) = A⎛⎝T12

ζ(1)n (kr)ζ(1)n (β)

⎞⎠

(A.132)

φ1,n(r) = A⎛⎝ζ(1)n (nkr)ζ(1)n (nβ)

+R11ζ(2)n (nkr)ζ(2)n (nβ)

⎞⎠

(A.133)

We can again match φ and its derivative φ′ at the boundary r = a to yield

the relation T12 = 1 +R11 and allows us to determine T12 and R11. Grandy Jr.

gives these coefficients for the magnetic case, j = 1 with his equations (5.18a)

- (5.18d) (Grandy Jr 2005, pg. 146).

It is also useful to see the algebraic derivation of these coefficients for both

the electric and magnetic multipoles, from Sn(β). We can rewrite (A.125) as

(A.136) making use of the notation of Nussenzveig (1969a), with j = 1,2:

[z] ≡ ln′ψn(z) =ψ′n(z)ψn(z)

(A.134)

[jz] ≡ ζ(j)′n (z)ζ(j)n (z)

(A.135)

Sn(β) = −ζ(2)n (β)ζ(1)n (β)

[2β] −mvj [mβ][1β] −mvj [nβ]

(A.136)

The second factor in (A.136) can be rewritten as:

[2β] −mvj [mβ][1β] −mvj [mβ]

= 1 − [1β] − [2β][1β] −mvj [2mβ]

[1β] −mvj [2mβ][1β] −mvj [mβ]

(A.137)

We can employ an identity (Grandy Jr.’s (5.11)) to transform the last factor

of (A.137), and further algebraic manipulation to rewrite the right hand side

309

of (A.137) as:

[2β] −mvj [mβ][1β] −mvj [mβ]

= −⎛⎝R22 (n,β) +

T21 (n,β)T12 (n,β)1 − ρ (n,β)

ζ(1)n (mβ)ζ(2)n (mβ)

⎞⎠

(A.138)

where:

ρ (n,β) ≡ −ζ(1)n (mβ)ζ(2)n (mβ)

[1β] −mvj [1mβ][1β] −mvj [2mβ]

(A.139)

(A.138) introduces the new functions for the spherical reflection and transmis-

sion coefficients:

R11 (n,β) ≡ζ(2)n (mβ)ζ(1)n (mβ)

ρ (n,β) (A.140)

R22 (n,β) ≡ −[2β] −mvj [2mβ][1β] −mvj [2mβ]

(A.141)

T12 (n,β) ≡[1mβ] − [2mβ]

[1β] −mvj [2mβ]= 1 +R11 (A.142)

T21 (n,β) ≡[1β] − [2β]

[1β] −mvj [2mβ]= 1 +R22 (A.143)

This allows us to rewrite (A.125) in terms of these coefficients:

Sn(β) =ζ(2)n (β)ζ(1)n (β)

⎛⎝R22 (n,β) +

ζ(1)n (mβ)ζ(2)n (mβ)

+ T21 (n,β)T12 (n,β)1 − ρ (n,β)

⎞⎠

(A.144)

We can introduce new forms of these coefficients by noticing that the ratios of

310

the Hankel functions in (A.144):

R11n (β) ≡ ζ

(1)n (mβ)ζ(2)n (mβ)

R11 (nβ) (A.145)

R22n (β) ≡ ζ

(2)n (β)ζ(1)n (β)

R22 (nβ) (A.146)

T 12n (β) ≡ ζ

(1)n (mβ)ζ(1)n (β)

T12 (nβ) (A.147)

T 21n (β) ≡ ζ

(2)n (β)

ζ(2)n (mβ)

T21 (nβ) (A.148)

which allows (A.144), the partial wave coefficients and the internal coefficients

to be written as follows:

Sn(β) = R22n + T

21n T

12n

1 −R11n

(A.149)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

an

bn

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭= 1

2(1 −R22

n − T21n T

12n

1 −R11n

) (A.150)

cn =T 21

1 −R11n

(A.151)

dn =T 12

1 −R11n

(A.152)

We can expand the denominator of (A.149) to obtain the Debye expansion

of the amplitude functions, Sjn. This can be expressed in terms of the spherical

coefficients as in (A.153), or in terms of the scattering functions as in (A.154):

Sjn = R22n + T 21

n

∞∑p=1

(R11n )p−1

T 12n (A.153)

Sj (β, θ) = Sj,0 (β,0) +P

∑p=1

Sj,p (β, θ) + remainder (A.154)

The first term in each case, R22n and Sj,0, is associated with direct reflection

from the surface and the pth term is associated with transmission after (p − 1)

311

internal reflections at the surface.

A.4 Poisson Sum

The Poisson sum formula is given in (A.155). This allows for an infinite sum

over a real function to be transformed into an integral over complex variables.

∞∑n=0

ψ (Λ, r) =∞∑

m=−∞(−1)m

∞

∫0

ψ (Λ, r) e2imπΛ dΛ (A.155)

where ψ is a certain function, Λ = n + 12 , n is the order of expansion of the

scattering functions, and r is a position vector.

A.5 Unified CAM Approach

Applying the modified Watson transform to the third term of the Debye ex-

pansion allows us to obtain a contour integral. We extend to the complex

plane so that we can deform the path to make the integration easier. In doing

so, we are able to “concentrate dominant contributions to the integrals into the

neighborhood of a small number of points in the λ plane, that will be referred

to as critical points. The main type of critical points are saddle points, that can

be real or complex, and complex poles such as the Regge poles” (Nussenzveig

1992, pg. 62). In this section I will set out the mathematics of the Watson

transformed third Debye term and the mathematics of the Chester-Friedman-

Ursell (CFU) method for resolving the collision between two saddle points.

The concept of poles is outlined in the final section.

312

A.5.1 Third Debye Term3

Based on Nussenzveig’s work, Adam offers a version of the Debye expansion

for the total scattering function:

f (β, θ) = f0 (β, θ) +∞∑p=1

fp (β, θ) (A.156)

fp (β, θ) = −i

β

∞∑n=−∞

(−1)n∞

∫0

U (λ,β) [γ (λ,β)]p−1Pλ− 1

2(cos θ) e2inπλλdλ

(A.157)

where fp is for p ≥ 1. U (λ,β) and γ (λ,β) in fp are given in (A.158) and

(A.159) respectively.4

U (λ,β) = T21 (λ,β)ζ(1)λ (α) ζ(2)λ (β)ζ(2)λ (α) ζ(1)λ (β)

T12 (λ,β) = U (−λ,β) (A.158)

γj =⎡⎢⎢⎢⎣ζ(1)n (α)ζ(2)n (α)

⎤⎥⎥⎥⎦Rj

11 (A.159)

From applying the modified Watson transformation to each individual term

of the Debye expansion, we can write the third term as:

f2 (β, θ) = −i

β

∞∑

m=−∞(−1)m

∞

∫0

γ (λ,β)U (λ,β)Pλ− 12e2imπλλdλ (A.160)

After rearrangement and a shift in the path of integration to above the real

axis, (A.160) can be written as:

f2 (β, θ) = −1

2β

∞−iε

∫−∞−iε

γUPλ− 12(cos θ) eiπλ λ

cosπλdλ (A.161)

3The derivation here is summarised from Adam (2002), who bases his work on Nussen-zveig (1969b).

4The spherical coefficients should be the same between Adam (2002), Grandy Jr (2005)and Nussenzveig (1992); any difference will be due to a difference in notation or rearrangingof terms.

313

for ε > 0, due to the function being odd. This integral can be decomposed into

(A.162), where f±2,0 (β, θ) is given in (A.163) and f±2,r (β, θ) is given in (A.164).

f2 (β, θ) = f+2,0 + f2,r = f−2,0 − f2,r (A.162)

f±2,0 (β, θ) = ±i

β

∞∓iε

∫−∞±iε

γUQ(2)λ− 1

2

(cos θ) dλ (A.163)

f2,r (β, θ) =1

2β

∞+iε

∫−∞−iε

γUe2iπλ λ

cosπλdλ (A.164)

Above, Q(2)λ− 1

2

is a Legendre function of the second kind. It can be shown

that there will always be some neighbourhood of the imaginary axis where the

integrand in (A.163) will diverge to infinity. This is true for any value of θ,

and so there is no domain of θ-values for which f±2,0 and hence f2 (β, θ) can be

reduced to a pure residue series. i.e. We need to use the saddle points.

A.5.2 CFU Method for Saddle Points

The rainbow is described as the collision between two saddle points. The

typical approach to finding the contribution to an integral from saddle points

is the method of ‘steepest descent’. This method fails, however, when the

range of two saddle points overlap. de Bruijin gives an informal definition of

the range of saddle points: “if ζ is a saddle point of the function ψ, then the

range of ζ is a circular neighbourhood of ζ, consisting of all z-values which are

such that ∣ψ′′(ζ) (z − ζ)2∣ is not very large” (de Bruijn 1958, pg. 91).

The principle behind the CFU method is to alter the exponent involved in

a general complex integral from a Gaussian type to an exact cubic.

Steepest Descent

Consider the general complex integral (A.165). The integral is over some path

in the w plane, where κ is an asymptotic expansion parameter and is large

314

and positive, and ε is an independent parameter. If this integral is dominated

by a single saddle point w = w(ε), around which f and g are regular, one can

approximate F (κ, ε) as f (w, ε) in this neighbourhood, where f ′′w denotes the

second derivative with respect to w.

F (κ, ε) = ∫ g(w)e[κf(w,ε)] dw (A.165)

f(w, ε) ≈ f(w, ε) + 1

2f ′′w(w, ε)(w − w)2 (A.166)

Choosing the steepest descent through the saddle point w alters the integral

to a Gaussian type. A change of variables allows us to transform the exponent

into an act Gaussian. This allows an asymptotic expansion of the integral to

be found by adopting a power series expansion of g around w and integrating

term by term.

The method of steepest descent allows for the contribution of saddle points

to be established. Provided that the range of two saddle points do not overlap,

then their contributions can be added independently. If their ranges do overlap,

then the method of steepest descent fails and we require the CFU method. This

overlap occurs in the rainbow region.

CFU Method: In General

The basic idea behind the CFU method is to transform the exponent of (A.166)

into an exact cubic through a change of variables, e.g. (A.167). This transforms

the two saddle points, w′ and w′′ into ±ζ 12 (ε). f(w, ε) is then substituted back

into (A.165), allowing for a suitable expansion of the integrand to obtain a

suitable equation for F (κ, ε).

f(w, ε) = 1

3u3 − ζ(ε)u +A(ε) (A.167)

315

CFU Method: The Rainbow

The CFU method applied to the rainbow requires an adjusted form of (A.160):

f2,g (β, θ) = −i

β

σ2∞

∫−σ1∞

ρUQ(2)λ− 1

2

(cos θ)λdλ (A.168)

where:

Q(2)λ− 1

2

= e2iπλQ(1)λ− 1

2

(cos θ) − ieiπλPλ− 12(− cos θ) (A.169)

Here the path of integration from (A.163) has been deformed into the path of

deepest descent, denoted by the change in limits of the integral to (−σ1∞, σ2∞).

This entails moving across the poles. This integral may be rewritten as:

f2,g = 2e−iπ4 N ( 2β

π sin θ)

12

F (β, θ) (A.170)

F (β, θ) =σ′2∞

∫−σ′1∞

g(w1)e2βf(w1,θ) dw1 (A.171)

f (w1, θ) = i [2N cosw2 − cosw1 + (2w2 −w1 −π − θ

2) sinw1] (A.172)

g (w1) = (sinw1)12 cos2w1 cosw2 [

cosw1 −N cosw2

(cosw1 +N cosw2)3 ] [1 +O(β−1)]

(A.173)

Above, equations (A.165) and (A.166) set out the general form of the equations

needed for the method of steepest descent. The specific versions of these

equations for the rainbow are equations (A.171) and (A.172) respectively.

The rest of the mathematics are too complex to include here. Further

details can be found in Nussenzveig (1969b) and Adam (2002). Nussenzveig

(1992) provides a summary of the derivation, which finished with equation

(A.174). This equation provides the form of dominant contribution to the

316

third Debye term in the rainbow region:

Sj2(β, θ) ≈ β76 e[2βA(ε)] [cj0(ε) + β−1cj1(ε) + . . . ]Ai [(2β)

23 ζ(ε)] (A.174)

+ [dj0(ε) + β−1dj1(ε) + . . . ]β−13Ai′ [(2β) 2

3 ζ(ε)]

A.6 Poles5

[A function] is analytic in a region of the complex plane if it has a (unique)

derivative at every point in the region. The statement ‘f(z) is analytic at a

point z = a’ means that f(z) has a derivative at every point inside some small

circle about z = a.

A regular point of f(z) is a point at which f(z) is analytic. A singular

point or singularity of f(z) is a point at which f(z) is not analytic. It is called

an isolated singular point if f(z) is analytic everywhere else inside some small

circle about the singular point.

Laurent’s Theorem:

Let C1 and C2 be two circles with centre at z0. Let f(z) be analytic in

the region R between the circles. Then f(z) can be expanded in a series

of the form

f(z) = a0 + a1(z − z0) + a2(z − z0)2 + ... + b1z − z0

+ b2(z − z0)2

+ ...

(A.175)

convergent on R. Such a series is called a Laurent series. The ‘b’ series

in (A.175) is called the principle part of the Laurent series.

If all the b’s are zero, f(z) is analytic at z = z0, and we call z0 a regular

point. If bn ≠ 0, but all the b’s after bn are zero, f(z) is said to have a pole of

order n at z = z0. If n = 1, we say that f(z) has a simple pole. If there are an5This entire section is reproduced directly from Boas (2005), pg. 667-670, and pg. 680.

317

infinite number of b’s different from zero, f(z) has an essential singularity at

z = z0. The coefficient b1 of 1z−z0 is called the residue of f(z) at z = z0.

318

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